arXiv:math/0201175v1 [math.AG] 18 Jan 2002 · such case E+ is a G-torsor for the e´tale topology...

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ALMOST RING THEORY

OFER GABBER AND LORENZO RAMERO

fifth release

January 17, 2018

Ofer GabberI.H.E.S.Le Bois-Marie35, route de ChartresF-91440 Bures-sur-Yvettee-mail address:[email protected]

Lorenzo RameroUniversite de Bordeaux ILaboratoire de Mathematiques351, cours de la LiberationF-33405 Talence Cedexe-mail address:[email protected] page:http://www.math.u-bordeaux.fr/∼ramero

A preliminary version of this manuscript was prepared in 1997, while the second author was supported by theIHES.

1

http://arxiv.org/abs/math/0201175v1

http://www.math.u-bordeaux.fr/~ramero

2 OFER GABBER AND LORENZO RAMERO

CONTENTS

1. Introduction 32. Homological theory 82.1. Some ring-theoretic preliminaries 82.2. Almost categories 102.3. Uniform spaces of almost modules 162.4. Almost homological algebra 212.5. Almost homotopical algebra 283. Almost ring theory 363.1. Flat, unramified and etale morphisms 363.2. Lifting theorems 383.3. Descent 433.4. Behaviour of etale morphisms under Frobenius 514. Fine study of almost projective modules 594.1. Almost traces 594.2. Endomorphisms ofGm. 654.3. Modules of almost finite rank 704.4. Localisation in the flat site 754.5. Construction of quotients by flat equivalence relations 825. Valuation theory 885.1. Ordered groups and valuations 885.2. Basic ramification theory 955.3. Algebraic extensions 995.4. Logarithmic differentials 1065.5. Transcendental extensions 1105.6. Deeply ramified extensions 1176. Analytic geometry 1236.1. Derived completion functor 1236.2. Cotangent complex for formal schemes and adic spaces 1286.3. Analytic geometry over a deeply ramified base 1336.4. Semicontinuity of the discriminant 1377. Appendix 1437.1. 143References 148Index 150

ALMOST RING THEORY 3

Le bruit des vagues etait encore plus paresseux, plusetale qu’a midi. C’etait le meme soleil, la memelumiere sur le meme sable qui se prolongeait ici.

A. Camus -L’ etranger

1. INTRODUCTION

From a pragmatic standpoint, one can describe the theory of almost rings as a useful tool forperforming calculations of Galois cohomology groups. Indeed, this is the main application ofFaltings’ “almost purity theorem”, which is the technical heart of [24].

Though almost ring theory is developed here as an independent branch of mathematics,stretching somewhere in between commutative algebra and category theory, the original ap-plications to Galois cohomology still provide the main motivation and influence largely theevolution of the subject.

It is therefore not inappropriate to introduce the present work by reviewing briefly the mainideas behind these calculations. Let us consider first a complete discretely valued fieldK ofzero characteristic, with perfect residue field of characteristic p > 0, and letK+ be the ring ofintegers ofK. The valuationv of K extends uniquely to any algebraic extension, and we wantto normalize the value group in such a way thatv(p) = 1 in every such extension.

Let E be a finite Galois extension ofK, with Galois groupG. Typically, one is given adiscreteE+[G]-moduleM (such that theG-action onM is semilinear, that is, compatiblewith theG-action onE+), and is interested in studying the reduced Galois cohomology H i :=

H i(G,M) (for i ∈ Z). (Recall thatH i(M) agrees with Galois cohomologyRiΓGM for i > 0,with Galois homology fori < −1, and fori = 0 it equalsMG/TrE/K(M), theG-invariantsdivided by the image of the trace map).

In such a situation, the scalar multiplication mapE+⊗ZM → M induces natural cup productpairingsH i ⊗Z H

j → H i+j. Especially, the action of(E+)G = K+ on H i factors throughK+/TrE/K(E

+); in other words, the image ofE+ under the trace map annihilates the reducedGalois cohomology.

If now the extensionE is tamely ramifiedoverK, thenTrE/K(E+) = K+, so theH i vanishfor all i ∈ Z. Even sharper results can be achieved when the extension isunramified. Indeed, insuch caseE+ is aG-torsor for the etale topology ofK+, hence, some basic descent theory tellsus that the natural map

E+ ⊗K+ RΓGM →M [0]

is an isomorphism in the derived category of the category ofE+[G]-modules (where we havedenoted byM [0] the complex consisting ofM placed in degree zero).

In Tate’s paper [48] there occurs a variant of the above situation : instead of the finite ex-tensionE one considers the algebraic closureKa of K, so thatG is the absolute Galois groupof K, and the discreteG-moduleM is replaced by thetopologicalmoduleC(χ), whereCis thep-adic completion ofKa, whose naturalG-action we “twist” by a continuous characterχ : G → K×. Then the relevantH• is thecontinuousGalois cohomologyH•cont(G,C(χ)),which is defined in general as the homology of a complex of continuous cochains. Under thepresent assumptions,H i can be computed by the formula:

H icont(G,C) := (lim

←−n

H i(G,Ka+(χ)⊗Z Z/pnZ))⊗Z Q.

Let nowK∞ be a totally ramified Galois extension with Galois groupH isomorphic toZp. Taterealized that, for cohomological purposes, the extensionK∞ plays the role of a maximal totallyramified Galois extension ofK. More precisely, letL be any finite extension ofK, and setLn := L ·Kn, whereKn is the subfield ofK∞ fixed byHpn ≃ pn ·Zp. The extensionKn ⊂ Ln


is unramified if and only if the different idealDn := DL+n /K

+n

equalsL+n . In case this fails,

the valuationv(δn) of a generatorδn of Dn will be a strictly positive rational number, giving aquantitative measure for the ramification of the extension.With this notation, [48,§3.2, Prop.9]reads

limn→∞

v(δn) = 0(1.0.1)

(indeed,v(δn) approaches zero about as fast asp−n). In this sense, one can say that the extensionK∞ ⊂ L∞ := L ·K∞ is almost unramified. One immediate consequence is that the maximalidealm ofK+

∞ is contained inTrL∞/K∞(L+∞). If, additionally,L is a Galois extension ofK, we

can consider the subgroup

G∞ := Gal(L∞/K∞) ⊂ Gal(L/K)

and the foregoing implies thatm annihilatesH icont(G∞,M), for everyi > 0, and every topolog-

icalL+∞[G∞]-moduleM . More precisely, the homology of the cone of the natural morphism

L+∞ ⊗K+

∞RΓG∞M →M [0](1.0.2)

is annihilated bym in all degrees,i.e. it is almost zero. Equivalently, one says that the maps onhomology induced by (1.0.2) arealmost isomorphismsin all degrees.

Tate goes on to apply these cohomological vanishings to the study of p-divisible groups;in turns, this study enables him to establish a comparison between the etale and the Hodgecohomology of an abelian scheme overK+, which has become the prototype for all subsequentinvestigation ofp-adic Hodge theory.

A first generalization of (1.0.1) can be found in the work [26]by Fresnel and Matignon; oneinteresting aspect of this work is that it does away with any consideration of local class fieldtheory (which was used to get the main estimates in [48]); instead, Fresnel and Matignon writea general extensionL as a tower of monogenic subextensions, whose structure is sufficientlywell understood to allow a direct and very explicit analysis. The main tool in [26] is a notionof different idealDE+/K+ for a possibly infinite algebraic field extensionK ⊂ E; then theextensionK∞ considered in [48] is replaced by any extensionE ofK such thatDE+/K+ = (0),and (1.0.1) is generalized by the claim that, for every finiteextensionE ⊂ F , DF+/E+ = F+.

In some sense, the arguments of [26] anticipate those used byFaltings in the first few para-graphs of his fundamental article [23]. There we find, first ofall, a further extension of (1.0.1):the residue field ofK is now not necessarily perfect, instead one assumes only that it admits afinite p-basis; then the relevantK∞ is an extension whose residue field is perfect, and whosevalue group isp-divisible. This generalization paves the way to the almostpurity theorem, ofwhich it represents the one-dimensional case. In order to state and prove the higher dimen-sional case, Faltings invents the method of “almost etale extensions”, and indeed sketches in afew pages a whole program of “almost commutative algebra”, with the aim of transposing tothe almost context as much as possible of the classical theory. So, for instance, ifA is a givenK+∞-algebra, andM is anA-module, one says thatM is almost flatif, for everyA-moduleN ,

the natural map of complexes

ML

⊗A N →M ⊗A N [0]

induces almost isomorphisms on homology in all degrees. Similarly, M is almost projectiveifthe same holds for the map of complexesHomA(M,N)[0]→ RHomA(M,N). Then, accordingto [23], a mapA → B of K+

∞-algebras is calledalmostetale if B is almost projective as anA-module and as aB ⊗A B-module (moreover,B is required to bealmost finitely generated:the discussion of finiteness conditions in almost ring theory is a rather subtle business, and wededicate the better part of section 2.3 to its clarification).

With this new language, the almost purity theorem should be better described as an almostversion of Abhyankar’s lemma, valid for morphismsA → B of K+-algebras that are etale in


characteristic zero and possibly wildly ramified on the locus of positive characteristic. Theactual statement goes as follows. Suppose thatA admits global etale coordinates, that is,there exists an etale mapK+[T±11 , ..., T±1d ] → A; whereas in the tamely ramified case a fi-nite cyclic ramified base changeA → A′ suffices to kill all ramification, the infinite extensionA∞ := A[T

±1/p∞

1 , ..., T±1/p∞

d ] ⊗K+ K+∞ is required in the wildly ramified case, to killalmost

all ramification, which means that the normalizationB∞ of A∞ ⊗A B is almost etale overA∞.Faltings has proposed two distinct strategies for the proofof his theorem : the first one, pre-

sented in [23], consists in adapting Grothendieck’s proof of Zariski-Nagata’s purity1; a more re-cent one ([24]) uses the action of Frobenius on some local cohomology modules, and is actuallyvalid under more general assumptions (one does not require the existence of etale coordinates,but only a weaker semi-stable reduction hypothesis on the special fibre).

As a corollary, one deduces cohomological vanishings generalizing the foregoing : indeed,suppose that the extension of fraction fieldsFrac(A) ⊂ Frac(B) is Galois with groupG; then,granting almost purity,B∞ is an “almostG-torsor” overA∞, therefore, for anyB∞[G]-moduleM , the natural map of complexesB∞ ⊗A∞ RΓGM → M [0] induces almost isomorphismson homology. Finally, these results can be used (together with a lot of hard work) to deducecomparison theorems betweenp-adic etale cohomology and deRham (or other kinds of) coho-mology, for arbitrary smooth projective varieties overK. This method can even be extendedto treat cohomology with not necessarily constant coefficients (see [24]), thereby providing themost comprehensive approach top-adic Hodge theory found so far.

The purpose of our text is to fully work out the foundations of“almost commutative algebra”outlined by Faltings; in the process we generalize and simplify considerably the theory, and alsoextend it in directions that were not explored in [23], [24].

Before passing to a more detailed presentation, we want to conclude this general review bymentioning the work of Coates and Greenberg [14], in which the notion ofdeeply ramifiedextensionof a local field is introduced, and applied to the study ofp-divisible groups attachedto abelian varieties defined over suchp-adic fields. Essentially, section 2 of [14] rediscovers theresults of Fresnel and Matignon, although via a different route, closer to the original treatmentof Tate in [48]. In particular, an algebraic extensionE of K is deeply ramified if and onlyif DE+/K+ = (0) according to the terminology of [26]. We adopt Coates and Greenberg’sterminology for our section 5.6, and we give some complements which were not observed in[14] (propositions 5.6.2 and 5.6.6). Furthermore, we extend the definition of deeply ramifiedextension to include valued fields of arbitrary rank.

It turns out that most of almost ring theory can be built up satisfactorily from a very slim andgeneral set of assumptions: our basic setup, introduced in section 2.1, consists of a ringV andan idealm ⊂ V such thatm = m2; starting from (2.5.2.5.14) we also assume thatm ⊗V m isa flatV -module : simple considerations show this to be a natural hypothesis, often verified inpractice.

TheV -modules killed bym are the objects of a (full) Serre subcategoryΣ of the categoryV -Mod of all V -modules, and the quotient categoryV a-Mod := V -Mod/Σ is an abeliancategory which we call the category ofalmostV -modules. It is easy to check that the usualtensor product ofV -modules descends to a bifunctor⊗ on almostV -modules, so thatV a-Mod

is a monoidal abelian category in a natural way. Then analmost ring is just an almostV -moduleA endowed with a “multiplication” morphismA ⊗ A → A satisfying certain naturalaxioms. Together with the obvious morphisms, these gadgetsform a categoryV a-Alg. Givenany almostV -algebraA, one can then define the notion ofA-module andA-algebra, just likefor usual rings. The purpose of the game is to reconstruct in this new framework as much aspossible (and useful) of classical linear and commutative algebra. The first useful observation

1At the time of writing, there are still some obscure points inthis proof


is that the localization functorV -Mod→ V a-Mod admits both left and right adjoints. Takentogether, these functors exhibit the kind of exactness properties that one associates to openembeddings of topoi, perhaps a hint of some deeper geometrical structure, still to be unearthed.Essentially, this is the same as the ideology informing Deligne’s paper [15], which sets out todevelop algebraic geometry in the context of abstract tannakian categories. We could also claiman even earlier ancestry, in that some of the leading motifs resonating throughout our text, canbe traced as far back as Gabriel’s memoir [27] “Des categories abeliennes”.

After these generalities, we treat in section 2.3 the question of finiteness conditions for almostmodules. LetA denote an almost algebra, fixed for the rest of this introduction. It is certainlypossible to define as usual a notion of finitely generatedA-module, however this turns out tobe too restrictive a class for applications. The main idea here is to define a uniform structureon the set of equivalence classes ofA-modules ; then we will say that anA-module isalmostfinitely generatedif its isomorphism class lies in the topological closure of the subspace offinitely generatedA-modules. Similarly we definealmost finitely presentedA-modules. Theuniform structure also comes handy when we want to constructoperators on almost modules :if one can show that the operator in question is uniformly continuous on a classC of almostmodules, then its definition extends right away by continuity to the topological closureC of C .This is exemplified by the construction of the (almost) Fitting ideals forA-modules, at the endof section 2.3.

In section 2.4 we introduce the basic toolkit of homologicalalgebra, beginning with thenotion of flat almost module, which poses no problem, since wedo have a tensor product in ourcategory. The notion of projectivity is more subtle : it turns out that the category ofA-modulesusually doesnot have enough projectives. The useful notion isalmost projectivity: simply oneuses the standard definition, except that the role of theHom functor is played by the internalalHom functor. The scarcity of projectives should not be regardedas surprising or pathological:it is quite analogous to the lack of enough projective objects in the category of quasi-coherentOX-modules on a non-affine schemeX.

Section 2.5 introduces the cotangent complex of a morphism of almost rings, which is put touse in the following chapter, to study infinitesimal liftings of almost algebras.

With chapter 3 we begin in earnest the study of almost commutative algebra: the classes offlat, unramified and etale morphisms are defined, together with some variants. In section 3.2we derive the infinitesimal lifting theorems for etale algebras (theorem 3.2.17) and for almostprojective modules (theorem 3.2.24).

Next we turn to study some cases of non-flat descent; when we specialize to usual rings, werecover known theorems (of course, standard commutative algebra is a particular case of almostring theory). But if the result is not new, the argument is : indeed, we believe that our treatment,even when specialized to usual rings, improves upon the method found in the literature.

The last section of chapter 3 calls on stage the Frobenius endomorphism of an almost algebraof positive characteristic. The main results are invariance of etale morphisms under pull-backby Frobenius maps (theorem 3.4.13) and theorem 3.4.28, which can be interpreted as a puritytheorem. Perhaps the most remarkable aspect of the latter result is how cheap it is : in positivecharacteristic, the availability of the Frobenius map allows for a quick and easy proof. Philo-sophically, this proof is not too far removed from the methoddevised by Faltings for his morerecent proof of purity in mixed characteristic.

An alternative title for chapter 4 could have been “Everything you can do with traces”. Rightat the outset we find the definition of the trace map of an almostprojective almost finitelygeneratedA-module. The whole purpose of the chapter is to showcase the versatility of thisconstruction, a real swiss-knife of almost linear algebra.For instance, we apply it to characterizeetale morphisms (theorem 4.1.14); more generally, it is used to define the different ideal of analmost finiteA-algebra. In section 4.3 it is employed in an essential way tostudy the important


class ofA-modules offinite rank, i.e. those almost projectiveA-modulesP such thatΛiAP = 0for sufficiently largei ∈ N. A rather complete and satisfactory description is achieved forsuchA-modules (proposition 4.3.27). This is further generalized in theorem 4.3.28, to arbitraryA-modules so called ofalmost finite rank(see definition 4.3.9(ii)). The interest of the latterclass is that it contains basically all the almost projective modules found in nature; indeed, wecannot produce a single example of an almost projective module that is almost finitely generatedbut has not almost finite rank (but we suspect that they do exist). In any case, almost finitelygenerated modules whose rank is not almost finite are certainly rather weird beasts : some clueabout their looks can be gained by analyzing the structure ofinvertiblemodules : we do this atthe end of section 4.4.

The other main construction of chapter 4 is thesplitting algebraof an almost projectivemodule, introduced in section 4.4 : with its aid we show thatA-modules of finite rank arelocally free in the flat topology ofA. It should be clear that this is a very pleasant and usefulculmination for our study of almost projective modules; we put it to use right away in thefollowing section 4.5, where we show that an etale groupoidof almost finite rank over thecategory of “affine almost schemes” (more prosaically, the opposite of the category of almostalgebras) is universally effective, that is, it admits a good quotient, as in the classical algebro-geometric setting.

Chapter 5 is an extended detour into valuation theory. The first two sections contain nothingnew, and are only meant to gather in a single place some usefulmaterial that is known to experts,but for which satisfactory references are hard to find. The main theme of sections 5.3 through5.5 is the study of the cotangent complex of an extension of valuation rings. To give a sampleof our results, suppose thatk is a perfect field, and letW be a valuation ring containingk; thenwe show thatΩW/k is a torsion-freeW -module. Notice that this assertion would be an easyconsequence of the existence of resolution of singularities for k-schemes; our methods enableus to prove it unconditionally, as well as several other statements and variants for logarithmicdifferentials. Almost rings make two appearances in this chapter: first, they are needed to definea different ideal for a finite separable extension of arbitrary valued fields of rank one; second,they are used in proposition 5.6.2 to characterize deeply ramified valued fields of rank one.

In the final chapter we bring into the picturep-adic analytic geometry and formal schemes.The main result is a kind of weak purity statement valid for affinoid varieties over a deeplyramified valued field of rank one (theorem 6.3.15). The occurence of analytic geometry inpurity issues (and inp-adic Hodge theory at large) is rather natural; indeed, the literature onthe subject is littered with indications of the relevance ofanalytic varieties (and already in [48],Tate explicitly asked for ap-adic Hodge theory in the framework of rigid varieties). In anycase, our theorem is much weaker than Faltings’ and does not yield by itself the kind of Galoiscohomology vanishings that are required to deduce comparison theorems for the cohomology ofalgebraic varieties; we explain more precisely the currentstatus of the question in (6.4.6.4.26).The chapter is rounded off with the sketch of a theory of the analytic cotangent complex forformal schemes and for Zariski-Riemann spaces (the latter seen as special cases of Huber’s adicspaces from [32]).

AcknowledgementsThe second author is very much indebted to Gerd Faltings for many patient explanations onthe method of almost etale extensions. Next he would like toacknowledge several interesting discussions withIoannis Emmanouil. He is also much obliged to Pierre Deligne, for a useful list of critical remarks. Finally, heowes a special thank to Roberto Ferretti, who has read the first tentative versions of this work, has corrected manyslips and has made several valuable suggestions.


2. HOMOLOGICAL THEORY

2.1. Some ring-theoretic preliminaries. Unless otherwise stated, every ring is commutativewith unit. This section collects some results of general nature that will be used throughout thiswork.

2.1.2.1.1. Our basic setup consists of a fixed base ringV containing an idealm such thatm2 = m. Starting from (2.5.2.5.14), we will also assume thatm := m⊗V m is a flatV -module.

Example 2.1.2. (i) The main example is given by a non-discrete valuation ring (V, | · |) of rankone; in this casem will be the maximal ideal.

(ii) Takem := V . This is the “classical limit”. In this case almost ring theory reduces to usualring theory. Thus, all the discussion that follows specialises to, and sometimes gives alternativeproofs for, statements about rings and their modules.

2.1.2.1.3. LetM be a givenV -module. We say thatM is almost zeroif m·M = 0. A mapφ ofV -modules is analmost isomorphismif bothKer(φ) andCoker(φ) are almost zeroV -modules.

Remark 2.1.4. (i) It is easy to check that aV -moduleM is almost zero if and only ifm⊗VM =0. Similarly, a mapM → N of V -modules is an almost isomorphism if and only if the inducedmapm⊗V M → m⊗V N is an isomorphism. Notice also that, ifm is flat, thenm ≃ m.

(ii) Let V → W be a ring homomorphism. For aV -moduleM setMW := W ⊗V M . Wehave an exact sequence

0→ K → mW → m ·W → 0(2.1.5)

whereK := TorV1 (V/m,W ) is an almost zeroW -module. By (i) it follows thatm ⊗V K ≃(m ·W )⊗W K ≃ 0. Then, applyingmW ⊗W − and−⊗W (m ·W ) to (2.1.5) we derive

mW ⊗W mW ≃ mW ⊗W (m ·W ) ≃ (m ·W )⊗W (m ·W )

i.e. mW ≃ (m ·W ) . In particular, ifm is a flatV -module, thenmW is a flatW -module. Thismeans that our basic assumptions on the pair(V,m) are stable under arbitrary base extension.Notice that the flatness ofm does not imply the flatness ofm ·W . This partly explains why weinsist thatm, rather thanm, be flat.

2.1.2.1.6. Before moving on, we want to analyze in some detail how our basic assumptionsrelate to certain other natural conditions that can be postulated on the pair(V,m). Indeed, let usconsider the following two hypotheses :(A) m = m2 andm is a filtered colimit of principal ideals.(B) m = m2 and, for all integersk > 1, thek-th powers of elements ofm generatem.

Clearly (A) implies (B). Less obvious is the following result.

Proposition 2.1.7. (i) (A) implies thatm is flat.

(ii) If m is flat then (B) holds.

Proof. Suppose that (A) holds, so thatm = colimα∈I

V xα, whereI is a directed set parametrizing

elementsxα ∈ m (andα ≤ β ⇔ V xα ⊂ V xβ). For anyα ∈ I we have natural isomorphisms

V xα ≃ V/AnnV (xα) ≃ (V xα)⊗V (V xα).(2.1.8)

Forα ≤ β, let jαβ : V xα → V xβ be the imbedding; we have a commutative diagram

Vµz2 //

πα

V

πβ

(V xα)⊗V (V xα)jαβ⊗jαβ // (V xβ)⊗V (V xβ)


wherez ∈ V is such thatxα = z ·xβ, µz2 is multiplication byz2 andπα is the projection inducedby (2.1.8) (and similarly forπβ). Sincem = m2, for all α ∈ I we can findβ such thatxα is amultiple ofx2β. Sayxα = y ·x2β; then we can takez := y ·xβ, soz2 is a multiple ofxα and in theabove diagramKer(πα) ⊂ Ker(µz2). Hence one can define a mapλαβ : (V xα)⊗V (V xα)→ Vsuch thatπβ λαβ = jαβ ⊗ jαβ andλαβ πα = µz2. It now follows that for everyV -moduleN , the induced morphismTorV1 (N, (V xα) ⊗V (V xα)) → TorV1 (N, (V xβ) ⊗V (V xβ)) is thezero map. Taking the colimit we derive thatm is flat. This shows (i). In order to show (ii) weconsider, for any prime numberp, the following condition

(∗p) m/p ·m is generated (as aV -module) by thep-th powers of its elements.

Clearly (B) implies (∗p) for all p. In fact we have :

Claim2.1.9. (B) holds if and only if (∗p) holds for every primep.

Proof of the claim:Suppose that (∗p) holds for every primep. The polarization identity

k! · x1 · x2 · ... · xk =∑

I⊂1,2,...,k

(−1)k−|I| ·

(∑

i∈I

xi

)k

shows that ifN :=∑

x∈mV xk thenk! · m ⊂ N . To prove thatN = m it then suffices to show

that for every primep dividing k! we havem = p · m +N . Let φ : V/p · V → V/p · V be theFrobenius (x 7→ xp); we can denote by(V/p · V )φ the ringV/p · V seen as aV/p · V -algebravia the homomorphismφ. Also setφ∗M := M ⊗V/p·V (V/p · V )φ for a V/p · V -moduleM .Then the mapφ∗(m/p ·m)→ (m/p ·m) (defined by raising top-th power) is surjective by (∗p).Hence so is(φr)∗(m/p ·m)→ (m/p ·m) for everyr > 0, which says thatm = p ·m+N whenk = pr, hence for everyk.

Next recall (see [2, Exp. XVII 5.5.2]) that, ifM is aV -module, the module of symmetrictensorsTSk(M) is defined as(⊗kVM)Sk , the invariants under the natural action of the symmetricgroupSk on⊗kVM . We have a natural mapΓk(M)→ TSk(M) that is an isomorphism whenMis flat (seeloc. cit. 5.5.2.5; hereΓk denotes thek-th graded piece of the divided power algebra).

Claim2.1.10. The groupSk acts trivially on⊗kVm and the mapm ⊗V m → m (x ⊗ y ⊗ z 7→x⊗ yz) is an isomorphism.

Proof of the claim:The first statement is reduced to the case of transpositions and tok = 2.There we can compute :x ⊗ yz = xy ⊗ z = y ⊗ xz = yz ⊗ x. For the second statement notethat the imbeddingm → V is an almost isomorphism, and apply remark 2.1.4(i).

Suppose now thatm is flat and pick a primep. ThenSp acts trivially on⊗pV m. Hence

Γp(m) ≃ ⊗pV m ≃ m.(2.1.11)

But Γp(m) is spanned as aV -module by the productsγi1(x1) · ... · γik(xk) (wherexi ∈ m and∑j ij = p). Under the isomorphism (2.1.11) these elements map to

(p

i1,...,ik

)· xi11 · ... · x

ikk ; but

such an element vanishes inm/p·m unlessik = p for somek. Thereforem/p·m is generated byp-th powers, so the same is true form/p ·m, and by the above, (B) holds, which shows (ii).

Proposition 2.1.12.Suppose thatm is countably generated as aV -module. Then we have :

(i) m is countably presented as aV -module;(ii) if m is a flatV -module, then it is of homological dimension≤ 1.


Proof. Let (εi)i∈I be a countable generating family ofm. Thenεi ⊗ εj generatem andεi · εj ·(εk⊗εl) = εk ·εl · (εi⊗εj) for all i, j, k, l ∈ I. For everyi ∈ I, we can writeεi =

∑j xijεj, for

certainxij ∈ m. LetF be theV -module defined by generators(eij)i,j∈I , subject to the relations:

εi · εj · ekl = εk · εl · eij eik =∑

j

xijejk for all i, j, k, l ∈ I.

We get an epimorphismπ : F → m by eij 7→ εi ⊗ εj . The relations imply that, ifx :=∑k,l yklekl ∈ Ker(π), thenεi · εj · x = 0, som · Ker(π) = 0. Whencem ⊗V Ker(π) = 0 and

1m⊗V π is an isomorphism. We consider the diagram

m⊗V F∼ //

φ

m⊗V m

ψ

F

π // m

whereφ andψ are induced by scalar multiplication. We already know thatψ is an isomorphism,and sinceF = m · F , we see thatφ is an epimorphism, soπ is an isomorphism, which shows(i). Now (ii) follows from (i), since it is well-known that a flat countably presented module is ofhomological dimension≤ 1 (see [38, Ch.I, Th.3.2] and the discussion in [42, pp.49-50]).

2.2. Almost categories. If C is a category, andX, Y two objects ofC , we will usually denoteby HomC (X, Y ) the set of morphisms inC from X to Y and by1X the identity morphismof X. Moreover we denote byC o the opposite category ofC and bys.C the category ofsimplicial objects overC , that is, functors∆o → C , where∆ is the category whose objects arethe ordered sets[n] := 0, ..., n for each integern ≥ 0 and where a morphismφ : [p] → [q]is a non-decreasing map. A morphismf : X → Y in s.C is a sequence of morphismsf[n] : X [n] → Y [n], n ≥ 0 such that the obvious diagrams commute. We can imbedC in s.Cby sending each objectX to the “constant” objects.X such thats.X [n] = X for all n ≥ 0 ands.X [φ] = 1X for all morphismsφ in ∆.

2.2.2.2.1. IfC is an abelian category,D(C ) will denote the derived category ofC . As usualwe have also the full subcategoriesD+(C ),D−(C ) of complexes of objects ofC that are exactfor sufficiently large negative (resp. positive) degree. IfR is a ring, the category ofR-modules(resp.R-algebras) will be denoted byR-Mod (resp.R-Alg). Most of the times we will writeHomR(M,N) instead ofHomR-Mod(M,N).

We denote bySet the category of sets. The symbolN denotes the set of non-negative inte-gers; in particular0 ∈ N.

2.2.2.2.2. The full subcategoryΣ of V -Mod consisting of allV -modules that are almostisomorphic to0 is clearly a Serre subcategory and hence we can form the quotient categoryV -Mod/Σ. There is a localization functor

V -Mod→ V -Mod/Σ M 7→Ma

that takes aV -moduleM to the same module, seen as an object ofV -Mod/Σ. In particular,we have the objectV a associated toV ; it seems therefore natural to use the notationV a-Mod

for the categoryV -Mod/Σ, and an object ofV a-Mod will be indifferently referred to as “aV a-module” or “an almostV -module”. In case we need to stress the dependance on the idealm, we can write(V,m)a-Mod.

Since the almost isomorphisms form a multiplicative system(seee.g. [50, Exerc.10.3.2]), itis possible to describe the morphisms inV a-Mod via a calculus of fractions, as follows. LetV -al.Iso be the category that has the same objects asV -Mod, but such thatHomV -al.Iso(M,N)consists of all almost isomorphismsM → N . If M is any object ofV -al.Iso we write(V -al.Iso/M) for the category of objects ofV -al.Iso overM (i.e. morphismsφ : X → M).


If φi : Xi → M (i = 1, 2) are two objects of(V -al.Iso/M) thenHom(V -al.Iso/M)(φ1, φ2)consists of all morphismsψ : X1 → X2 in V -al.Iso such thatφ1 = φ2 ψ. For any twoV -modulesM,N we define a functorFN : (V -al.Iso/M)o → V -Mod by associating to anobjectφ : P → M the V -moduleHomV (P,N) and to a morphismα : P → Q the mapHomV (Q,N)→ HomV (P,N) : β 7→ β α. Then we have

HomV a-Mod(Ma, Na) = colim

(V -al.Iso/M)oFN .(2.2.3)

However, formula (2.2.3) can be simplified considerably by remarking that for anyV -moduleM , the natural morphismm ⊗V M → M is an initial object of(V -al.Iso/M). Indeed, letφ : N →M be an almost isomorphism; the diagram

m⊗V N∼ //

m⊗V M

N

φ // M

(cp. remark 2.1.4(i)) allows one to define a morphismψ : m ⊗V M → N overM . Weneed to show thatψ is unique. But ifψ1, ψ2 : m ⊗V M → N are two maps overM , thenIm(ψ1−ψ2) ⊂ Ker(φ) is almost zero, henceIm(ψ1−ψ2) = 0, sincem⊗V M = m · (m⊗V M).Consequently, (2.2.3) boils down to

HomV a-Mod(Ma, Na) = HomV (m⊗V M,N).(2.2.4)

In particularHomV a-Mod(M,N) has a natural structure ofV -module for any twoV a-modulesM,N , i.e.HomV a-Mod(−,−) is a bifunctor that takes values in the categoryV -Mod.

2.2.2.2.5. One checks easily (for instance using (2.2.4)) that the usual tensor product inducesa bifunctor− ⊗V − on almostV -modules, which, in the jargon of [16] makes ofV a-Mod anabelian tensor category. Then analmostV -algebra is just a commutative unitary monoid inthe tensor categoryV a-Mod. Let us recall what this means. Quite generally, let(C ,⊗, U) beany abelian tensor category, so that⊗ : C × C → C is a biadditive functor,U is the identityobject ofC (see [16, p.105]) and for any two objectsM andN in C we have a “commutativityconstraint” (i.e. a functorial isomorphismθM |N : M ⊗ N → N ⊗M that “switches the twofactors”) and a functorial isomorphismνM : U ⊗M → M . Then aC -monoidA is an objectof C endowed with a morphismµA : A ⊗ A → A (the “multiplication” ofA) satisfying theassociativity condition

µA (1A ⊗ µA) = µA (µA ⊗ 1A).

We say thatA is unitary if additionallyA is endowed with a “unit morphism”1A : U → Asatisfying the (left and right) unit property :

µA (1A ⊗ 1A) = νA µA (1A ⊗ 1A) θA|U = µA (1A ⊗ 1A).

Finally A is commutativeif µA = µA θA|A (to be rigorous, in all of the above one shouldindicate the associativity constraints, which we have omitted : see [16]). A commutative unitarymonoid will also be simply called analgebra. With the morphisms defined in the obvious way,theC -monoids form a category; furthermore, given aC -monoidA, a leftA-moduleis an objectM of C endowed with a morphismσM/A : A ⊗M → M such thatσM/A (1A ⊗ σM/A) =σM/A (µA ⊗ 1M). Similarly one defines rightA-modules andA-bimodules. In the case ofbimodules we have left and right morphismsσM,l : A⊗M →M , σM,r :M ⊗A→ M and oneimposes that they “commute”,i.e. that

σM,r (σM,l ⊗ 1A) = σM,l (1A ⊗ σM,r).

Clearly the (left resp. right)A-modules (and theA-bimodules) form an additive category withA-linear morphismsdefined as one expects. One defines the notion of a submodule asan


equivalence class of monomorphismsN →M such that the compositionA⊗N → A⊗M →M factors throughN . Especially, atwo-sided idealof A is anA-sub-bimoduleI → A. Forgiven submodulesI, J ofA one denotesIJ := Im(I⊗J → A⊗A

µA→ A). For anA-moduleM ,theannihilatorAnnA(M) of M is the largest idealj : I → A of A such that the composition

I ⊗Mj⊗1M−→ A⊗M

σM/A−→ M is the zero morphism.

2.2.2.2.6. Iff : M → N is a morphism of leftA-modules, thenKer(f) exists in the under-lying abelian categoryC and one checks easily that it has a unique structure of leftA-modulewhich makes it a submodule ofM . If moreover⊗ is right exactwhen either argument is fixed,then alsoCoker(f) has a uniqueA-module structure for whichN → Coker(f) isA-linear. Inthis case the category of leftA-modules is abelian. Similarly, ifA is a unitaryC -monoid, thenone defines the notion ofunitary left A-module by requiring thatσM/A (1A ⊗ 1M) = νM andthese form an abelian category when⊗ is right exact.

2.2.2.2.7. Specialising to our case we obtain the categoryV a-Alg of almostV -algebras and,for every almostV -algebraA, the categoryA-Mod of unitary leftA-modules. Clearly thelocalization functor restricts to a functorV -Alg → V a-Alg and for anyV -algebraR we havea localization functorR-Mod→ Ra-Mod.

Next, if A is an almostV -algebra, we can define the categoryA-Alg of A-algebras. Itconsists of all the morphismsA→ B of almostV -algebras.

2.2.2.2.8. Let again(C ,⊗, U) be any abelian tensor category. By [16, p.119], the endomor-phism ringEndC (U) ofU is commutative. For any objectM of C , denoteM∗ = HomC (U,M);thenM 7→ M∗ defines a functorC → EndC (U)-Mod. Moreover, ifA is a C -monoid,A∗is an associativeEndC (U)-algebra, with multiplication given as follows. Fora, b ∈ A∗ leta · b := µA (a ⊗ b) ν−1U . Similarly, if M is anA-module,M∗ is anA∗-module in a nat-ural way, and in this way we obtain a functor fromA-modules andA-linear morphisms toA∗-modules andA∗-linear maps. Using [16, Prop.1.3], one can also check thatEndC (U) = U∗asEndC (U)-algebras, whereU is viewed as aC -monoid usingνU .

2.2.2.2.9. All this applies especially to our categories ofalmost modules and almost algebras: in this case we callM 7→ M∗ the functor of almost elements. So, ifM is an almost module,an almost element ofM is just an honest element ofM∗. Using (2.2.4) one can show easily thatfor everyV -moduleM the natural mapM → (Ma)∗ is an almost isomorphism.

2.2.2.2.10. LetA be aV a-algebra. For any twoA-modulesM,N , the setHomA-Mod(M,N)has a natural structure ofA∗-module and we obtain an internal Hom functor by letting

alHomA(M,N) = HomA-Mod(M,N)a.

This is the functor ofalmost homomorphismsfromM toN .

2.2.2.2.11. For anyA-moduleM we have also a functor of tensor productM ⊗A − on A-modules which, in view of the following proposition 2.2.13 can be shown to be a left adjoint tothe functoralHomA(M,−). It can be defined asM ⊗A N := (M∗ ⊗A∗ N∗)

a.With this tensor product,A-Mod is an abelian tensor category as well, andA-Alg could

also be described as the category of (A-Mod)-algebras. Under this equivalence, a morphismφ : A→ B of almostV -algebras becomes the unit morphism1B : A→ B of the correspondingmonoid. We will sometimes drop the subscript and write simply 1.

Remark 2.2.12. Let V → W be a map of base rings,W taken with the extended idealm ·W .ThenW a is an almostV -algebra so we have defined the categoryW a-Mod using base ringVand the category(W,m ·W )a-Mod using baseW . One shows easily that they are equivalent:


we have an obvious functor(W,m·W )a-Mod→W a-Mod and an essential inverse is providedbyM 7→M∗. Similar base comparison statements hold for the categories of almost algebras.

Proposition 2.2.13.LetA be aV a-algebra,R a V -algebra.

(i) There is a natural isomorphismA ≃ Aa∗ of almostV -algebras.(ii) The functorM 7→ M∗ fromRa-Mod toR-Mod (resp. fromRa-Alg toR-Alg) is right

adjoint to the localization functorR-Mod→ Ra-Mod (resp.R-Alg→ Ra-Alg).(iii) The counit of the adjunctionMa

∗ → M is a natural isomorphism from the composition ofthe two functors to the identity functor1Ra-Mod (resp.1Ra-Alg).

Proof. (i) has already been remarked. We show (ii). In light of remark 2.2.12 (applied withW = R) we can assume thatV = R. LetM be aV -module andN an almostV -module; wehave natural bijections

HomV a-Mod(Ma, N)≃ HomV a-Mod(M

a, (N∗)a) ≃ HomV (m⊗V M,N∗)

≃ HomV (M,HomV (m, N∗)) ≃ HomV (M,HomV a-Mod(V, (N∗)a))

≃ HomV (M,N∗)

which proves (ii). Now (iii) follows by inspecting the proofof (ii), or by [27, Ch.III Prop.3].

Remark 2.2.14. (i) Let M1,M2 be twoA-modules. By proposition 2.2.13(iii) it is clear thata morphismφ : M1 → M2 of A-modules is uniquely determined by the induced morphismM1∗ → M2∗. On this basis, we will very often define morphisms ofA-modules (orA-algebras)by saying how they act on almost elements.

(ii) It is a bit tricky to deal with preimages of almost elements under morphisms: for instance,if φ :M1 →M2 is an epimorphism (by which we mean thatCoker(φ) ≃ 0) andm2 ∈M2∗, thenit is not true in general that we can find an almost elementm1 ∈ M1∗ such thatφ∗(m1) = m2.What remains true is that for arbitraryε ∈ m we can findm1 such thatφ∗(m1) = ε ·m2.

(iii) The existence of the right adjointM 7→ M∗ follows also directly from [27, Chap.III§3Cor.1 or Chap.V§2].

Corollary 2.2.15. The categoriesA-Mod andA-Alg are both complete and cocomplete.

Proof. We recall that the categoriesA∗-Mod andA∗-Alg are both complete and cocomplete.Now let I be any small indexing category andM : I → A-Mod be any functor. Denote byM∗ : I → A∗-Mod the composed functori 7→M(i)∗. We claim thatcolim

IM = (colim

IM∗)

a.

The proof is an easy application of proposition 2.2.13(iii). A similar argument also works forlimits and for the categoryA-Alg.

2.2.2.2.16. For anyV a-algebraA, The abelian categoryA-Mod satisfies axiom (AB5) (seee.g. [50, §A.4]) and it has a generator, namely the objectA itself. It then follows by a generalresult thatA-Mod has enough injectives.

Corollary 2.2.17. The functorM 7→ M∗ fromRa-Mod to R-Mod sends injectives to injec-tives and injective envelopes to injective envelopes.

Proof. The functorM 7→ M∗ is right adjoint to an exact functor, hence it preserves injectives.Now, let J be an injective envelope ofM ; to show thatJ∗ is an injective envelope ofM∗, itsuffices to show thatJ∗ is an essential extension ofM∗. However, ifN ⊂ J∗ andN ∩M∗ = 0,thenNa ∩M = 0, hencem ·N = 0, butJ∗ does not containm-torsion, thusN = 0.


2.2.2.2.18. Note that the essential image ofM 7→ M∗ is closed under limits. Next recall thatthe forgetful functorA∗-Alg→ Set (resp.A∗-Mod→ Set) has a left adjointA∗[−] : Set→A∗-Alg (resp.A(−) : Set→ A∗-Mod) that assigns to a setS the freeA∗-algebraA∗[S] (resp.the freeA∗-moduleA(S)

∗ ) generated byS. If S is any set, it is natural to writeA[S] (resp.A(S))for theA-algebra(A∗[S])a (resp. for theA-module(A(S)

∗ )a. This yields a left adjoint, calledthefreeA-algebrafunctorSet→ A-Alg (resp. thefreeA-modulefunctorSet→ A-Mod) tothe “forgetful” functorA-Alg→ Set (resp.A-Mod→ Set) B 7→ B∗.

2.2.2.2.19. Now letR be anyV -algebra; we want to construct a left adjoint to the localisationfunctorR-Mod→ Ra-Mod. For a givenRa-moduleM , let

M! := m⊗V (M∗).(2.2.20)

We have the natural map (unit of adjunction)R→ Ra∗, so that we can viewM! as anR-module.

Proposition 2.2.21.LetR be aV -algebra.

(i) The functorRa-Mod→ R-Mod defined by(2.2.20)is left adjoint to localisation.(ii) The unit of the adjunctionM → Ma

! is a natural isomorphism from the identity functor1Ra-Mod to the composition of the two functors.

Proof. (i) follows easily from (2.2.4) and (ii) follows easily from(i).

Corollary 2.2.22. Suppose thatm is a flatV -module. Then we have :

(i) the functorM 7→M! is exact;(ii) the localisation functorR-Mod→ Ra-Mod sends injectives to injectives.

Proof. By proposition 2.2.21 it follows thatM 7→ M! is right exact. To show that it is also leftexact whenm is a flatV -module, it suffices to remark thatM 7→ M∗ is left exact. Now, by (i),the functorM 7→Ma is right adjoint to an exact functor, so (ii) is clear.

2.2.2.2.23. LetB be anyA-algebra. The multiplication onB∗ is inherited byB!, which istherefore a non-unital ring in a natural way. We endow theV -moduleV ⊕ B! with the ringstructure determined by the rule:(v, b) · (v′, b′) := (v · v′, v · b′ + v′ · b+ b · b′) for all v, v′ ∈ Vandb, b′ ∈ B!. ThenV ⊕B! is a (unital) ring. We notice that theV -submodule generated by allthe elements of the form(x · y,−x⊗ y⊗ 1) (for arbitraryx, y ∈ m) forms an idealI of V ⊕B!.SetB!! := (V ⊕ B!)/I. Thus we have a sequence ofV -modules

0→ m→ V ⊕ B! → B!! → 0(2.2.24)

which in general is only right exact.

Definition 2.2.25. We say thatB is anexactA-algebraif the sequence (2.2.24) is exact.

Remark 2.2.26. Notice that if m∼→ m (e.g. whenm is flat), then allA-algebras are exact.

In the general case, ifB is anyA-algebra, thenV a × B is always exact. Indeed, we have(V a ×B)∗ ≃ V a

∗ × B∗ and, by remark 2.1.4(i),m⊗V V a∗ ≃ m.

Clearly we have a natural isomorphismB ≃ Ba!!.

Proposition 2.2.27.The functorB 7→ B!! is left adjoint to the localisation functorA!!-Alg →A-Alg.

Proof. Let B be anA-algebra,C anA!!-algebra andφ : B → Ca a morphism ofA-algebras.By proposition 2.2.21 we obtain a naturalA∗-linear morphismB! → C. Together with thestructure morphismV → C this yields a mapφ : V ⊕ B! → C which is easily seen to be aring homomorphism. It is equally clear that the idealI defined above is mapped to zero byφ,


hence the latter factors through a map ofA!!-algebrasB!! → C. Conversely, such a map inducesa morphism ofA-algebrasB → Ca just by taking localisation. It is easy to check that the twoprocedures are inverse to each other, which shows the assertion.

Remark 2.2.28. The functor of almost elements commutes with arbitrary limits, because allright adjoints do. It does not in general commute with colimits, not even with arbitrary infinitedirect sums. Dually, the functorsM 7→ M! andB 7→ B!! commute with all colimits. Inparticular, the latter commutes with tensor products.


2.3. Uniform spaces of almost modules.

2.3.2.3.1. LetA be aV a-algebra. For any cardinal numberc, we let Mc(A) be the set ofisomorphism classes ofA-modules which admit a set of generators of cardinality≤ c. In thefollowing we fix some (very) large infinite cardinalityω, and suppose that the isomorphismclasses of all ourA-modules lie inMω(A). The choice ofω is required to avoid set-theoreticalinconsistencies, but it is immaterial for our purposes, so we will henceforth just writeM (A)instead ofMω(A).

Definition 2.3.2. LetA be aV a-algebra andM anA-module.

(i) We define a uniform structure on the setIA(M) of A-submodules ofM , as follows.For every finitely generated idealm0 ⊂ m, the subset ofIA(M) × IA(M) given byEM(m0) := (M0,M1) | m0 · M0 ⊂ M1 andm0 · M1 ⊂ M0 is an entourage for theuniform structure, and the subsets of this kind form a fundamental system of entourages.

(ii) We define a uniform structure onM (A) as follows. For every finitely generated idealm0 ⊂ m and every integern ≥ 0 we define the entourageEM (m0) ⊂ M (A) ×M (A),which consists of all pairs ofA-modules(M,M ′) such that there exist a third moduleN and morphismsφ : N → M , ψ : N → M ′, such thatm0 annihilates the kerneland cokernel ofφ andψ. We declare that theEM (m0) form a fundamental system ofentourages for the uniform structure ofM (A).

Remark 2.3.3. Notice that the entourageEM (m0) can be defined equivalently by all the pairsof A-modules(M,M ′) such that there exists a third moduleL and morphismsφ′ : M → L,ψ : M ′ → L such thatm0 annihilates the kernel and cokernel ofφ andφ. Indeed, given apair (M,M ′) ∈ EM (m0), and a datum(N, φ, ψ) as in definition 2.3.2(ii) a datum(L, φ′, ψ′)satisfying the above condition is obtained from the push outdiagram

Nφ //

ψ

M

φ′

M ′

ψ′

// L.

(2.3.4)

Conversely, given a datum(L, φ′, ψ′), one obtains another diagram as (2.3.4), by lettingN bethe fibred product ofM andM ′ overL.

2.3.2.3.5. We will also need occasionally a notion of “Cauchy product” : let∏∞

n=0 In be aformal infinite product of idealsIn ⊂ A. We say that the formal productsatisfies the Cauchycondition(or briefly : is a Cauchy product) if, for every neighborhoodU of A in IA(A) thereexistsn0 ≥ 0 such that

∏n+pm=n Im ∈ U for all n ≥ n0 and allp ≥ 0.

Lemma 2.3.6. LetM be anA-module.

(i) IA(M) with the uniform structure of definition 2.3.2 is complete and separated.(ii) The following maps are uniformly continuous :

(a) IA(M)×IA(M)→ IA(M) : (M ′,M ′′) 7→M ′ ∩M ′′.(b) IA(M)×IA(M)→ IA(M) : (M ′,M ′′) 7→M ′ +M ′′.(c) IA(A)×IA(A)→ IA(A) : (I, J) 7→ I · J .

(iii) For anyA-linear morphismφ :M → N , the following maps are uniformly continuous:(a) IA(M)→ IA(N) : M ′ 7→ φ(M ′).(b) IA(N)→ IA(M) : N ′ 7→ φ−1(N ′).

Proof. (i) : The separation property is easily verified. We show thatIA(M) is complete.Therefore, suppose thatF is some Cauchy filter ofIA(M). Concretely, this means that for


every finitely generatedm0 ⊂ m, there existsF (m0) ∈ F such thatm0 · I ⊂ J for ev-ery I, J ∈ F (m0). Let L :=

⋃F∈F

(⋂I∈F I). We claim thatL is the limit of our filter.

Indeed, for a given finitely generatedm0 ⊂ m, we havem0 · I ⊂⋂J∈F (m0)

J , for everyI ∈ F (m0), whencem0 · I ⊂ L. On the other hand, ifI ∈ F ⊂ F (m0), we can write:m0 · L =

⋃F ′⊂F m0 · (

⋂J∈F ′ J) ⊂

⋃F ′⊂F (

⋂J∈F ′ m0 · J) ⊂

⋃F ′⊂F I = I (whereF ′ runs

over all the subsetsF ′ ∈ F such thatF ′ ⊂ F ). This shows that(L, I) ∈ EM(m0) wheneverI ∈ F (m0), which implies the claim. (ii) and (iii) are easy and will be left to the reader.

Remark 2.3.7. In general, the uniform spaceM (A) is not separated. In view of proposition3.2.26, a counterexample is provided by remark 3.2.25.

Lemma 2.3.8. Let φ : M → N be anA-linear morphism,B an A-algebra. The followingmaps are uniformly continuous :

(i) M (A)→ IA(A) : M 7→ AnnA(M).(ii) IA(M)×IA(N)→M (A) : (M ′, N ′) 7→ (φ(M ′) +N ′)/φ(M ′).(iii) M (A)×M (A)→M (A) : (M ′,M ′′) 7→ alHomA(M

′,M ′′).(iv) M (A)×M (A)→M (A) : (M ′,M ′′) 7→ M ′ ⊗AM ′′.(v) M (A)→M (B) : M 7→ B ⊗AM .(vi) M (A)→M (A) : M 7→ ΛrAM for anyr ≥ 0, provided(B) holds.

Proof. We show (iv) and leave the others to the reader. By symmetry, we reduce to verifyingthat, if (M ′,M ′′) ∈ EM (m0) andN is an arbitraryA-module, then(N ⊗A M ′, N ⊗A M ′′) ∈EM (m2

0). Then we can further assume that there is a morphismφ : M ′ → M ′′ with m0 ·Ker(φ) = m0 · Coker(φ) = 0. We factorφ! as an epimorphism followed by a monomorphism

M ′!φ1→ Im(φ!)

φ2→ M ′′! , and then we reduce to checking that the kernels and cokernels of both1N ⊗A φ1 and1N ⊗A φ2 are killed bym0. This is clear for1N ⊗A φ1, and it follows easily for1N ⊗A φ2 as well, by using the Tor sequences.

Definition 2.3.9. For a subsetS of a topological spaceT , letS denote the adherence ofS in T .LetM be anA-module.

(i) M is said to befinitely generatedif its isomorphism class lies in⋃n∈N Mn(A).

(ii) M is said to bealmost finitely generatedif its isomorphism class lies in⋃n∈N Mn(A).

(iii) M is said to beuniformly almost finitely generatedif there exists an integern ≥ 0 suchthat the isomorphism class ofM lies in Un(A) := Mn(A). Then we will say thatn is auniform boundfor M .

(iv) M is said to befinitely presentedif it is isomorphic to the cokernel of a morphism offree finitely generatedA-modules. We denote byFP(A) ⊂ M (A) the subset of theisomorphism classes of finitely presentedA-modules.

(v) M is almost finitely presentedif its isomorphism class lies inFP(A).

Remark 2.3.10. Under condition (A), anA-moduleM lies in Un(A) if and only if, for everyε ∈ m there exists anA-linear morphismAn →M whose cokernel is killed byε.

Proposition 2.3.11.LetM be anA-module.

(i) M is almost finitely generated if and only if for every finitely generated idealm0 ⊂ mthere exists a finitely generated submoduleM0 ⊂M such thatm0 ·M ⊂M0.

(ii) The following conditions are equivalent:(a) M is almost finitely presented.(b) for arbitrary ε, δ ∈ m there exist positive integersn = n(ε), m = m(ε) and a three

term complexAmψε→ An

φε→M with ε · Coker(φε) = 0 andδ ·Ker(φε) ⊂ Im(ψε).


(c) For every finitely generated idealm0 ⊂ m there is a complexAmψ→ An

φ→ M with

m0 · Coker(φ) = 0 andm0 ·Ker(φ) ⊂ Im(ψ).

Proof. (i): Let M be an almost finitely generatedA-module, andm0 ⊂ m a finitely generatedsubideal. Choose a finitely generated subidealm1 ⊂ m such thatm0 ⊂ m3

1; by hypothesis, thereexistA-modulesM ′ andM ′′, whereM ′′ is finitely generated, and morphismsf :M ′ →M , g :M ′ →M ′′ whose kernels and cokernels are annihilated bym1. We get morphismsm1⊗VM ′′ →Im(g) andm1 ⊗V Im(g)→ M ′, hence a composed morphismφ : m1 ⊗V m1 ⊗V M ′′ → M ′; itis easy to check thatCoker(f φ) is annihilated bym3

1, henceM0 := Im(f φ) will do.To show (ii) we will need the following :

Claim 2.3.12. Let F1 be a finitely generatedA-module and suppose that we are givena, b ∈ Vand a (not necessarily commutative) diagram

F1

p //

φ

M

F2

ψ

OO

q

>>

such thatq φ = a · p, p ψ = b · q. Let I ⊂ V be an ideal such thatKer(q) has a finitelygenerated submodule containingI · Ker(q). ThenKer(p) has a finitely generated submodulecontaininga · b · I ·Ker(p).

Proof of the claim: Let R be the submodule ofKer(q) given by the assumption. We haveIm(ψφ−a·b·1F1) ⊂ Ker(p) andψ(R) ⊂ Ker(p). We takeR1 := Im(ψφ−a·b·1F1)+ψ(R).Clearlyφ(Ker(p)) ⊂ Ker(q), soI ·φ(Ker(p)) ⊂ R, henceI ·ψφ(Ker(p)) ⊂ ψ(R) and finallya · b · I ·Ker(p) ⊂ R1.

Claim 2.3.13. If M satisfies condition (b) of the proposition, andφ : F → M is a morphismwith F ≃ An, then for every finitely generated idealm1 ⊂ m · AnnV (Coker(φ)) there is afinitely generated submodule ofKer(φ) containingm1 ·Ker(φ).

Proof of the claim:Now, letδ ∈ AnnV (Coker(φ)) andε1, ε2, ε3, ε4 ∈ m. By assumption there

is a complexArt→ As

q→ M with ε1 · Coker(q) = 0, ε2 · Ker(q) ⊂ Im(t). LettingF1 := F ,

F2 := As, a := ε1 · ε3, b := ε4 · δ, one checks easily thatψ andφ can be given such that all theassumptions of claim 2.3.12 are fulfilled. So, withI := ε2 ·V we see thatε1 ·ε2 ·ε3 ·ε4 ·δ ·Ker(φ)lies in a finitely generated submodule ofKer(φ). Butm1 is contained in an ideal generated byfinitely many such productsε1 · ε2 · ε3 · ε4 · δ.

Now, it is clear that (c) implies (a) and (b). To show that (b) implies (c), take a finitelygenerated idealm1 ⊂ m such thatm0 ⊂ m ·m1, pick a morphismφ : An → M whose cokernelis annihilated bym1, and apply claim 2.3.13. We show that (a) implies (c). For a given finitelygenerated subidealm0 ⊂ m, pick another finitely generatedm1 ⊂ m such thatm0 ⊂ m3

1; findmorphismsf :M ′ →M andg :M ′ → M ′′ whose kernels and cokernels are annihilated bym1,and such thatM ′′ is finitely presented. Letε := (ε1, ...εr) be a finite sequence of generators ofm1, and denote byK• := K•(ε) the Koszul complex ofV -modules associated to the sequenceε. Setm′1 := Coker(K2 → K1); we derive a natural surjection∂ : m′1 → m1 and, for everyi = 1, ..., r, mapsei : V → m′1 such that the compositions

m′1∂→ m1 → V

ei→ m′1 Vei→ m′1

∂→ m1 → V

are both scalar multiplication byεi. Hence, for everyV a-moduleN , the kernel and cokernel ofthe natural morphismm′1⊗V N → N are annihilated bym1. Let nowφ be as in the proof of (i);


notice that the diagram:m1 ⊗V m1 ⊗V M ′ //

M ′

m1 ⊗V m1 ⊗V M ′′ //

φ

77ooooooooooooo

M ′′

commutes. It follows that the composed morphism

m′1 ⊗V m′1 ⊗V M′′ → m1 ⊗V m1 ⊗V M

′′ φ→ M ′ → M

has kernel and cokernel annihilated bym31, so the claim follows.

The following proposition generalises a well-known characterization of finitely presentedmodules over usual rings.

Proposition 2.3.14.LetM be anA-module.

(i) M is almost finitely generated if and only if, for every filteredsystem(Nλ, φλµ) (indexedby a directed setΛ) the natural morphism

M : colimΛ

alHomA(M,Nλ)→ alHomA(M, colimΛ

Nλ)(2.3.15)

is a monomorphism.(ii) M is almost finitely presented if and only if for every filtered inductive sytem as above,

(2.3.15)is an isomorphism.

Proof. The “only if” part in (i) (resp. (ii)) is first checked whenM is finitely generated (resp.finitely presented) and then extended to the general case. Weleave the details to the reader andwe proceed to verify the “if” part. For (i), choose a setI and an epimorphismp : A(I) → M .Let Λ be the directed set of finite subsets ofI, ordered by inclusion. ForS ∈ Λ, let MS :=p(AS). Thencolim

Λ(M/MS) = 0, so the assumption givescolim

ΛalHomA(M,M/MS) = 0,

i.e. colimΛ

HomA(M,M/MS) = 0 is almost zero, so, for everyε ∈ m, the image ofε · 1Min the above colimit is0, i.e. there existsS ∈ Λ such thatε · M ⊂ MS, which proves thecontention. For (ii), we presentM as a filtered colimitcolim

ΛMλ, where eachMλ is finitely

presented (this can be donee.g.by taking such a presentation of theA∗-moduleM∗ and applyingN 7→ Na). The assumption of (ii) gives thatcolim

ΛHomA(M,Mλ) → HomA(M,M) is an

almost isomorphism, hence, for everyε ∈ m there isλ ∈ Λ andφε : M → Mλ such thatpλ φε = ε · 1M , wherepλ : Mλ → M is the natural morphism to the colimit. If such aφεexists forλ, then it exists for everyµ ≥ λ. Hence, ifm0 ⊂ m is a finitely generated subideal,saym0 =

∑kj V εj , then there existλ ∈ Λ andφi : M → Mλ such thatpλ φi = εi · 1M for

i = 1, ..., k. HenceIm(φipλ−εi·1Mλ) is contained inKer(pλ) and containsεi·Ker(pλ). Hence

Ker(pλ) has a finitely generated submoduleL containingm0 · Ker(pλ). Choose a presentationAm → An

π→ Mλ. Then one can liftm0 · L to a finitely generated submoduleL′ of An. Then

Ker(π)+L′ is a finitely generated submodule ofKer(pλ π) containingm20 ·Ker(pλ π). Since

we also havem0 ·Coker(pλπ) = 0 andm0 is arbitrary, the conclusion follows from proposition2.3.11.

Lemma 2.3.16.Let0→M ′ →M → M ′′ → 0 be an exact sequence ofA-modules. Then:

(i) If M ′,M ′′ are almost finitely generated (resp. presented) then so isM .(ii) If M is almost finitely presented, thenM ′′ is almost finitely presented if and only ifM ′ is

almost finitely generated.

Proof. These facts can be deduced from proposition 2.3.14 and remark 2.4.9(iii), or proveddirectly.


Lemma 2.3.17.Let Mn ; φn : Mn → Mn+1 | n ∈ N be a direct system ofA-modules andsuppose there exist sequencesεn | n ∈ N andδn | n ∈ N of ideals ofV such that

(i) limn→∞

εan = V a (for the uniform structure of definition 2.3.2) and∏∞

j=0 δj is a Cauchy

product (see(2.3.2.3.5);(ii) for all n ∈ N there exist integersN(n) and morphisms ofA-modulesψn : AN(n) → Mn

such thatεn · Coker(ψn) = 0;(iii) δn · Coker(φn) = 0 for all n ∈ N.

Thencolimn∈N

Mn is an almost finitely generatedA-module.

Proof. LetM := colimn∈N

Mn. For anyn ∈ N let an =⋂m≥0(

∏n+mj=n δj). Then lim

n→∞an = V . For

m > n setφn,m = φm ... φn+1 φn : Mn → Mm+1 and letφn,∞ : Mn → M be the naturalmorphism. An easy induction shows that

∏mj=n δj · Coker(φn,m) = 0 for all m > n ∈ N. Since

Coker(φn,∞) = colimm∈N

Coker(φn,m) we obtainan · Coker(φn,∞) = 0 for all n ∈ N. Therefore

εn · an · Coker(φn,∞ ψn) = 0 for all n ∈ N. Since limn→∞

εn · an = V , the claim follows.

In the remaining of this sectionwe assume that condition (B) of (2.1.2.1.6)holds. We wishto define the Fitting ideals of an arbitrary uniformly almostfinitely generatedA-moduleM .This will be achieved in two steps: first we will see how to define the Fitting ideals of a finitelygenerated module, then we will deal with the general case. Werefer to [38, Ch.XIX] for thedefinition of the Fitting idealsFi(M) of a finitely generated module over an arbitrary ringR.

Lemma 2.3.18.Let R be aV -algebra andM , N two finitely generatedR-modules with anisomorphism ofRa-modulesMa ≃ Na. ThenFi(M)a = Fi(N)a for everyi ≥ 0.

Proof. By the usual arguments, for everyε ∈ m we have morphismsα :M → N , β : N →Mwith kernels and cokernels killed byε2. Then we have:Fi(N) ⊃ Fi(Im(α)) · F0(Coker(α)). IfN is generated byk elements, then the same holds forCoker(α), whenceAnnR(Coker(α))k ⊂F0(Coker(α)), thereforeε2k · R ⊂ F0(Coker(α)), and consequentlyFi(N) ⊃ ε2k · Fi(Im(α)).SinceIm(α) is a quotient ofM , it is clear thatFi(M) ⊂ Fi(Im(α)), so finallyε2k · Fi(M) ⊂Fi(N). Arguing symmetrically withβ one hasε2k ·Fi(N) ⊂ Fi(M). Since we assume (B), theclaim follows.

2.3.2.3.19. LetM be a finitely generatedA-module. In light of lemma 2.3.18, the FittingidealsFi(M) are well defined as ideals inA.

Lemma 2.3.20.Letm0 ⊂ m be a finitely generated subideal andn ∈ N. Pick ε1, ..., εk ∈ msuch thatm0 ⊂ (ε3n1 , ..., ε

3nk ) and setm1 := (ε1, ..., εk). Then(Fi(M), Fi(M

′)) ∈ EA(m0) forevery(M,M ′) ∈ EM (m1) such thatM andM ′ are generated by at mostn of their almostelements.

Proof. Let M,M ′ be as in the lemma. By hypothesis, there exist anA-moduleN and mor-phismsφ : N → M andψ : N →M ′ such thatm1 annihilates the kernel and cokernel ofφ andψ. By symmetry, it suffices to show thatε3ni · Fi(M) ⊂ Fi(M

′) for everyi = 1, ..., k. Now, foreveryi ≤ k, the morphismM → M : x 7→ εi · x factors through a morphismα : M → φ(N),and similarly, scalar multiplication byεi onN factors through a morphismβ : φ(N) → N .Thenη := ψ β α : M → M ′ has kernel and cokernel annihilated byε3i . Pick finitelygeneratedA∗-modulesL ⊂ M∗, L′ ⊂ M ′∗ such thatLa = M andL′a = M ′. ReplacingL′

by L′ + η∗(L) we can assume thatη∗(L) ⊂ L′. ThenFi(M) = Fi(L)a, Fi(L′)a = Fi(M

′)andFi(L′) ⊃ Fi(L/(L ∩ Ker η∗)) · F0(L

′/(η∗L)). SinceL′/η∗(L) is generated by at mostn elements and is annihilated byε3i · m, we haveε3ni · m ⊂ F0(L

′/(η∗L)). FurthermoreFi(L/(L ∩Ker η∗)) ⊃ Fi(L), so the claim follows.


Proposition 2.3.21.For everyi, n ≥ 0, the mapFi : Mn(A) → IA(A) is uniformly continu-ous and therefore it extends uniquely to a uniformly continuous mapFi : Un(A)→ IA(A).

Proof. The uniform continuity follows readily from lemma 2.3.20. SinceIA(A) is complete, itfollows thatFi extends to the whole ofUn(A). Finally, the extension is unique becauseIA(A)is separated.

Definition 2.3.22. LetM be a uniformly almost finitely generatedA-module. We callFi(M)thei-th Fitting idealof M .

Proposition 2.3.23. (i) Let0→ M ′φ→ M

ψ→ M ′′ → 0 be a short exact sequence of uniformly

almost finitely generatedA-modules. Then:∑

j+k=i Fj(M′)·Fk(M ′′) ⊂ Fi(M) for everyi ≥ 0.

(ii) For every uniformly almost finitely generatedA-moduleM , anyA-algebraB and anyi ≥ 0 we haveFi(B ⊗AM) = Fi(M) · B.

Proof. (i): Let n be uniform bound forM andM ′; by remark 2.3.3 we can find, for every

subidealm0 ⊂ m, A-modulesM0, M ′0, L, L′ and morphismsMα→ L

β← M0,M ′

α′

→ L′β′

← M ′0whose kernels and cokernels are annihilated bym0, and such thatM0 andM ′0 are generated byn almost elements. LetN be defined by the push-out diagram

M ′φ //

α′

Mα // L

γ

L′

γ′ // N.

Furthermore setM ′1 := Im(γ′β ′ :M ′0 → N),M1 := Im((γβ)⊕(γ′β ′) :M0⊕M ′0 → N) andletM ′′1 be the cokernel of the induced monomorphismM ′1 → M1. We deduce a commutativediagram with short exact rows:

0 // M ′φ //

Mψ //

M ′′

// 0

0 // Im(γ′) // N // Coker(γ′) // 0

0 // M ′1

OO

// M1

OO

// M ′′1

OO

// 0.

(2.3.24)

One checks easily that the kernels and cokernels of all the vertical arrows in (2.3.24) are an-nihilated bym2

0, i.e. (M,M1), (M′,M ′1), (M

′′,M ′′1 ) ∈ EM (m20). Let x1, ..., xn ∈ M0∗ (resp.

x′1, ..., x′n ∈ M ′0∗) be a set of generators forM0 (resp. forM ′0). For everyi = 1, ..., n, let

zi := γ β(xi) andz′i := γ′ β ′(x′i). Let Q′ ⊂ N∗ (respQ ⊂ N∗) be theA∗-module gen-erated by thez′i (resp. and by thezi). It is clear that the bottom row of (2.3.24) is naturallyisomorphic to the short exact sequence(0 → Q′ → Q → Q/Q′ → 0)a. It is well-knownthat Fi(Q′) · Fj(Q/Q′) ⊂ Fi+j(Q) for every i, j ∈ N; by lemma 2.3.6(ii.c) and proposi-tion 2.3.21 all the operations under considerations are uniformly continuous, so we deduceFi(M

′) · Fj(M/M ′) ⊂ Fi+j(M), which is (i).(ii): since the identity is known for usual finitely generated modules over rings, the claim

follows easily from proposition 2.3.21 and lemma 2.3.8(v).

2.4. Almost homological algebra. In this section we fix an almostV -algebraA and we con-sider various constructions in the category ofA-modules.


2.4.2.4.1. By corollary 2.2.15 any inverse systemMn | n ∈ N ofA-modules has an (inverse)limit lim

n∈NM . As usual, we denote bylim1 the right derived functor of the inverse limit functor.

Notice that [50, Cor. 3.5.4] holds in the almost case since axiom (AB4*) holds inA-Mod (onthe other hand, it is not clear whether [50, Lemma 3.5.3] holds under (AB4*), since the proofuses elements).

Lemma 2.4.2. LetMn ; φn :Mn →Mn+1 | n ∈ N (resp.Nn ; ψn : Nn+1 → Nn | n ∈ N)be a direct (resp. inverse) system ofA-modules and morphisms andεn | n ∈ N a sequence ofideals ofV a converging toV a (for the uniform structure of definition 2.3.2).

(i) If εn ·Mn = 0 for all n ∈ N thencolimn∈N

Mn ≃ 0.

(ii) If εn ·Nn = 0 for all n ∈ N thenlimn∈N

Nn ≃ 0 ≃ limn∈N

1Nn.

(iii) If εn · Coker(ψn) = 0 for all n ∈ N and∏∞

j=0 εj is a Cauchy product, thenlimn∈N

1Nn ≃ 0.

Proof. (i) and (ii) : we remark only thatlimn∈N

1Nn ≃ limn∈N

1Nn+p for all p ∈ N and leave the details

to the reader. We prove (iii). From [50, Cor. 3.5.4] it follows easily that(limn∈N

1Nn∗)a ≃ lim

n∈N

1Nn.

It then suffices to show thatlimn∈N

1Nn∗ is almost zero. We haveε2n · Coker(ψn∗) = 0 and the

product∏∞

j=0(ε2j) is again a Cauchy product. Next letN ′n :=

⋂p≥0 Im(Nn+p∗ → Nn∗). If

Jn :=⋂p≥0(εn · εn+1 · ... · εn+p)2 thenJn · Nn∗ ⊂ N ′n and lim

n→∞Jan = V a. In view of (ii),

limn∈N

1Nn∗/N′n is almost zero, hence we reduce to showing thatlim

n∈N

1N ′n is almost zero. But

Jn+p+q ·N′n ⊂ Im(N ′n+p+q → N ′n) ⊂ Im(N ′n+p → N ′n)

for all n, p, q ∈ N. On the other hand, since the idealsJan converge toV a, we get⋃∞q=0m ·

Jn+p+q = m, hencem ·N ′n ⊂ Im(N ′n+p → N ′n) and finallym ·N ′n = m2 ·N ′n ⊂ Im(m ·N ′n+p →

m ·N ′n) which means thatm ·N ′n is a surjective inverse system, so itslim1 vanishes and the

result follows.

Example 2.4.3.Let (V,m) be as in example 2.1.2. Then every finitely generated ideal inV isprincipal, so in the situation of the lemma we can writeεj = (xj) for somexj ∈ V . Then thehypothesis in (iii) can be stated by saying that there existsc ∈ N such thatxj 6= 0 for all j ≥ cand the sequencen 7→

∏nj=c |xj| is Cauchy inΓ.

Definition 2.4.4. LetM be anA-module.

(i) We say thatM is flat (resp.faithfully flat) if the functorN 7→ M ⊗AN , from the categoryof A-modules to itself is exact (resp. exact and faithful).

(ii) We say thatM is almost projectiveif the functorN 7→ alHomA(M,N) is exact.

For euphonic reasons, we will use the expression ”almost finitely generated projective” to de-note anA-module which is almost projective and almost finitely generated. This conventiondoes not give rise to ambiguities, since we will never consider projective almost modules (in-deed, the categorical notion of projectivity is useless in the setting of almost ring theory).

Lemma 2.4.5. Let P be one of the properties : “flat”, “almost projective”, “almost finitelygenerated”, “almost finitely presented”. IfB is aP A-algebra, andM is aP B-module, thenM isP as anA-module.

Proof. Left to the reader.


2.4.2.4.6. LetR be aV -algebra andM a flat (resp. faithfully flat)R-module (in the usualsense, see [41, p.45]). ThenMa is a flat (resp. faithfully flat)Ra-module. Indeed, the functorM ⊗R− preserves the Serre subcategory of almost zero modules, so by general facts it inducesan exact functor on the localized categories (cp. [27, p.369]). For the faithfullness we have toshow that anR-moduleN is almost zero wheneverM ⊗RN is almost zero. However,M ⊗RNis almost zero⇔ M ⊗R (m ⊗V N) = 0⇔ m ⊗V N = 0⇔ N is almost zero. It is clear thatA-Mod has enough almost projective (resp. flat) objects.

2.4.2.4.7. LetR be aV -algebra. The localisation functor induces a functorG : D(R) →D(Ra) and, in view of corollary 2.2.22,M 7→ M! induces a functorF : D(Ra) → D(R). Wehave a natural isomorphismG F ≃ 1D(Ra) and a natural transformationF G → 1D(R).These satisfy the triangular identities of [40, p.83] soF is a left adjoint toG. If Σ denotes themultiplicative set of morphisms inD(R) which induce almost isomorphisms on the cohomologymodules, then the localised categoryΣ−1D(R) exists (seee.g.[50, Th.10.3.7]) and by the sameargument we get an equivalence of categoriesΣ−1D(R) ≃ D(Ra).

2.4.2.4.8. Given anA-moduleM , we can derive the functorsM ⊗A− (resp.alHomA(M,−),resp. alHomA(−,M)) by taking flat (resp. injective, resp. almost projective) resolutions :one remarks that bounded above exact complexes of flat (resp.almost projective)A-modulesare acyclic for the functorM ⊗A − (resp. alHomA(−,M)) (recall the standard argument: ifF• is a bounded above exact complex of flatA-modules, letΦ• be a flat resolution ofM ; thenTot(Φ•⊗AF•)→M⊗AF• is a quasi-isomorphism since it is so on rows, andTot(Φ•⊗AF•) isacyclic since its columns are; similarly, ifP• is a complex of almost projective objects, one con-siders the double complexalHomA(P•, J

•) whereJ• is an injective resolution ofM ; cp. [50,§2.7]); then one uses the construction detailed in [50, Th.10.5.9]. We denote byTorAi (M,−)(resp.alExtiA(M,−), resp.alExtiA(−,M)) the corresponding derived functors. IfA := Ra forsomeV -algebraR, we obtain easily natural isomorphismsTorRi (M,N)a ≃ TorAi (M

a, Na) forall R-modulesM,N . A similar result holds forExtiR(M,N).

Remark 2.4.9. (i) Clearly, anA-moduleM is flat (resp. almost projective) if and only ifTorAi (M,N) = 0 (resp. alExtiA(M,N) = 0) for all A-modulesN and all i > 0. In par-ticular, an almost projectiveA-module is flat, because for everyε ∈ m the scalar multiplicationby ε : M →M factors through a free module.

(ii) Let M,N be two flat (resp. almost projective)A-modules. ThenM ⊗A N is a flat (resp.almost projective)A-module and for anyA-algebraB, theB-moduleB ⊗A M is flat (resp.almost projective).

(iii) Resume the notation of proposition 2.3.14. IfM is almost finitely presented, then one hasalso that the natural morphismcolim

ΛalExt1A(M,Nλ) → alExt1A(M, colim

ΛNλ) is a monomor-

phism. This is deduced from proposition 2.3.14(ii), using the fact that(Nλ) can be injectedinto an inductive system(Jλ) of injectiveA-modules (e.g. Jλ = EHomA(Nλ,E), whereE is aninjective cogenerator forA-Mod), and by applyingalExt sequences.

Lemma 2.4.10.LetM be an almost finitely generatedA-module. ThenM is almost projectiveif and only if, for arbitraryε ∈ m, there existn(ε) ∈ N andA-linear morphisms

Muε−→ An(ε)

vε−→M(2.4.11)

such thatvε uε = ε · 1M .

Proof. Let morphisms as in (2.4.11) be given. Pick anyA-moduleN and apply the functoralExtiA(−, N) to (2.4.11) to get morphisms

alExtiA(M,N)→ alExtiA(An(ε), N)→ alExtiA(M,N)


whose composition is again the scalar multiplication byε; henceε · alExti(M,N) = 0 forall i > 0. Sinceε is arbitrary, it follows from remark 2.4.9(i) thatM is almost projective.Conversely, suppose thatM is almost projective; by hypothesis, for arbitraryε ∈ m we canfind n := n(ε) and a morphismφε : An → M such thatε · Coker(φε) = 0. Let Mε be the

image ofφε, so thatφε factors asAn(ε)ψε−→ Mε

jε−→ M . Also ε · 1M : M → M factors as

Mγε−→Mε

jε−→M. Since by hypothesisM is almost projective, the natural morphism induced

by ψε :

alHomA(M,An)ψ∗ε−→ alHomA(M,Mε)

is an epimorphism. Then for arbitraryδ ∈ m the morphismδ · γε is in the image ofψ∗ε , in otherwords, there exists anA-linear morphismuεδ :M → An such thatψε uεδ = δ · γε. If now wetakevεδ := φε, it is clear thatvεδ uεδ = ε · δ · 1M . This proves the claim.

Lemma 2.4.12.LetR be any ring,M anyR-module andC := Coker(φ : Rn → Rm) anyfinitely presented (left)R-module. LetC ′ := Coker(φ∗ : Rm → Rn) be the cokernel of thetranspose of the mapφ. Then there is a natural isomorphism

TorR1 (C′,M) ≃ HomR(C,M)/Im(HomR(C,R)⊗R M).

Proof. We have a spectral sequence :

E2ij := TorRi (Hj(Cone(φ

∗)),M)⇒ Hi+j(Cone(φ∗)⊗RM).

On the other hand we have also natural isomorphisms

Cone(φ∗)⊗RM ≃ HomR(Cone(φ), R)[1]⊗RM ≃ HomR(Cone(φ),M)[1].

Hence :E2

10 ≃ E∞10 ≃ H1(Cone(φ∗)⊗RM)/E∞01 ≃H

0(HomR(Cone(φ),M))/Im(E201)

≃ HomR(C,M)/Im(HomR(C,R)⊗R M)

which is the claim.

Proposition 2.4.13.LetA be aV a-algebra.

(i) Every almost finitely generated projectiveA-module is almost finitely presented.(ii) Every almost finitely presented flatA-module is almost projective.

Proof. (ii) : let M be such anA-module. Letε, δ ∈ m and pick a three term complex

Amψ−→ An

φ−→ M

such thatε ·Coker(φ) = δ ·Ker(φ)/Im(ψ) = 0. SetP := Coker(ψ∗); this is a finitely presentedA∗-module andφ∗ factors through a morphismφ∗ : P → M∗. Let γ ∈ m; from lemma 2.4.12we see thatγ · φ is the image of some element

∑nj=1 φj ⊗ mj ∈ HomA∗(P,A∗) ⊗A∗ M∗.

If we defineL := An∗ and v : P → L, w : L → M∗ by v(x) := (φ1(x), ..., φn(x)) andw(y1, ..., yn) :=

∑nj=1 yj ·mj, then clearlyγ ·φ = wv. LetK := Ker(φ∗). Thenδ ·Ka = 0 and

the mapδ ·1P a factors through a morphismσ : (P/K)a → P a. Similarly the mapε ·1M factorsthrough a morphismλ :M → (P/K)a. Letα := vaσλ :M → La andβ := wa : La →M .The reader can check thatβ α = ε · δ · γ · 1M . By lemma 2.4.10 the claim follows.

(i) : let P be such an almost finitely generated projectiveA-module. For any finitely gener-ated idealm0 ⊂ m pick a morphismφ : Ar → P such thatm0 ·Coker(φ) = 0. If ε1, ..., εk is a setof generators form0, a standard argument shows that, for anyi ≤ k, εi · 1P lifts to a morphismψi : P → Ar/Ker(φ); then, sinceP is almost projective,εjψi lifts to a morphismψij : P → Ar.Now claim 2.3.12 applies withF1 := Ar, F2 := M = P , p := φ, q := 1P andψ := ψij andshows thatKer(φ) has a finitely generated submoduleMij containingεi · εj ·Ker(φ). Then the


span of all suchMij is a finitely generated submodule ofKer(φ) containingm20 · Ker(φ). By

proposition 2.3.11(ii), the claim follows.

In general a flat almost finitely generatedA-module is not necessarily almost finitely pre-sented, but one can give the following criterion, which extends [12,§1, Exerc.13].

Proposition 2.4.14. If A→ B is a monomorphism ofV a-algebras andM is an almost finitelygenerated flatA-module such thatB⊗AM is almost finitely presented overB, thenM is almostfinitely presented overA.

Proof. Let m0 be a finitely generated subideal ofm and anA-linear morphismφ : An → Msuch thatm0⊗Coker(φ) = 0. By assumption and claim 2.3.13, we can find a finitely generatedB-submoduleR of Ker(1B ⊗A φ) such that

m20 ·Ker(1B ⊗A φ) ⊂ R.(2.4.15)

By a Tor sequence we havem0 · Coker(1B ⊗A Ker(φ)→ Ker(1B ⊗A φ)) = 0, hencem · m0 ·Coker(1B∗ ⊗A∗ Ker(φ)∗ → Ker(1B ⊗A φ)∗) = 0, therefore there exists a finitely generatedsubmoduleR0 of Ker(φ) such that

m20 · R ⊂ B · Im(R0 → Ker(1B ⊗A φ)).(2.4.16)

By lemma 2.4.12, for everyε ∈ m the morphismε · φ : An/R0 → M factors through amorphismψ : An/R0 → F , whereF is a finitely generated freeA-module. SinceF ⊂ B⊗AF ,we deduce easilyKer(An/R0 → Bn/B · R0) ⊂ Ker(ψ); on the other hand, by (2.4.15) and(2.4.16) we derivem4

0 · Ker(φ) ⊂ Ker(An/R0 → Bn/B · R0). Thusψ factors throughM ′ :=An/(R0 +m4

0 ·Ker(φ)). Clearlym40 ·Ker(M ′ →M) = m0 ·Coker(M ′ → M) = 0; hence, for

everyA-moduleN , the kernel of the induced morphism

alExt1A(M,N)→ alExt1A(M′, N)(2.4.17)

is annihilated bym50; however (2.4.17) factors throughalExt1A(F,N) = 0, thereforem5

0 ·alExt1A(M,N) = 0 for everyA-moduleN . This shows thatM is almost projective, whichis equivalent to the conclusion, in view of proposition 2.4.13(i).

Definition 2.4.18. LetM be anA-module.

(i) ThedualA-moduleof M is theA-moduleM∗ := alHomA(M,A).(ii) The evaluation morphismis the morphismevM/A :M ⊗AM∗ → A : m⊗ φ 7→ φ(m).(iii) The evaluation idealof M is the idealEM/A := Im(evM/A).(iv) We say thatM is reflexiveif the natural morphism

M → (M∗)∗ m 7→ (f 7→ f(m))(2.4.19)

is an isomorphism ofA-modules.(v) We say thatM is invertible if M ⊗AM∗ ≃ A.

Remark 2.4.20. Notice that ifB is anA-algebra andM anyB-module, then by “restrictionof scalars”M is also anA-module and the dualA-moduleM∗ has a natural structure ofB-module. This is defined by the rule(b · f)(m) := f(b · m) (b ∈ B∗, m ∈ M∗ andf ∈ M∗∗ ).With respect to this structure (2.4.19) becomes aB-linear morphism. Incidentally, notice thatthe two meanings of “M∗∗ ” coincide,i.e. (M∗)∗ ≃ (M∗)∗.


2.4.2.4.21. IfE, F andN areA-modules, there is a natural morphism :

E ⊗A alHomA(F,N)→ alHomA(F,E ⊗A N).(2.4.22)

Let P be anA-module. As a special case of (2.4.22) we have the morphism:

ωP/A : P ⊗A P∗ → EndA(P )

a := alHomA(P, P )

such thatωP/A(p⊗ φ)(q) := p · φ(q) for everyp, q ∈ P∗ andφ : P → A.

Proposition 2.4.23.LetP be an almost projectiveA-module.

(i) For every morphism of algebrasA→ B we haveEB⊗AP/B = EP/A · B.(ii) EP/A = E 2

P/A.(iii) P = 0 if and only ifEP/A = 0.(iv) P is faithfully flat if and only ifEP/A = A.

Proof. Pick an indexing setI large enough, and an epimorphismφ : F := A(I) → P . Foreveryi ∈ I we have the standard morphismsA

ei→ Fπi→ A such thatπi ej = δij · 1A and∑

i∈I ei πi = 1F . For everyx ∈ m chooseψx ∈ HomA(P, F ) such thatφ ψx = x · 1P .It is easy to check thatEP/A is generated by the almost elementsπi ψx φ ej (i, j ∈ I,x ∈ m). (i) follows already. For (iii), the “only if” is clear; ifEP/A = 0, thenψx φ = 0 for allx ∈ m, henceψx = 0 and thereforex · 1P = 0. Next, notice that, from (i) and (iii) we deriveP/(EP/A · P ) = 0, i.e. P = EP/A · P , so (ii) follows directly from the definition ofEP/A. SinceP is flat, to show (iv) we have only to verify that the functorM 7→ P ⊗AM is faithful. To thispurpose, it suffices to check thatP ⊗A (A/J) 6= 0 for every proper idealJ of A. This followseasily from (i) and (iii).

Lemma 2.4.24.LetE, F ,N be threeA-modules.

(i) The morphism(2.4.22)is an isomorphism in the following cases :(a) whenE is flat andF is almost finitely presented;(b) when eitherE or F is almost finitely generated projective;(c) whenF is almost projective andE is almost finitely presented;(d) whenE is almost projective andF is almost finitely generated.

(ii) The morphism(2.4.22)is a monomorphism in the following cases :(a) whenE is flat andF is almost finitely generated;(b) whenE is almost projective.

(iii) The morphism(2.4.22)is an epimorphism whenF is almost projective andE is almostfinitely generated.

Proof. If F ≃ A(I) for some finite setI, thenalHomA(F,N) ≃ N (I) and the claims areobvious. More generally, ifF is almost finitely generated projective, for anyε ∈ m there existsa finite setI := I(ε) and morphisms

Fuε−→ A(I) vε−→ F(2.4.25)

such thatvε uε = ε · 1F . We apply the natural transformation

E ⊗A alHomA(−, N)→ alHomA(−, E ⊗A N)

to (2.4.25) : an easy diagram chase allows then to conclude that the kernel and cokernel of(2.4.22) are killed byε. As ε is arbitrary, it follows that (2.4.22) is an isomorphism in this case.An analogous argument works whenE is almost finitely generated projective, so we get (i.b).If F is only almost projective, then we still have morphisms of the type (2.4.25), but nowI(ε)is no longer necessarily finite. However, the cokernels of the induced morphisms1E ⊗ uε andalHomA(vε, E ⊗A N) are still annihilated byε. Hence, to show (iii) (resp. (i.c)) it sufficesto consider the case whenF is free andE is almost finitely generated (resp. presented). By


passing to almost elements, we can further reduce to the analogous question for usual ringsand modules, and by the usual juggling we can even replaceE by a finitely generated (resp.presented)A∗-module andF by a freeA∗-module. This case is easily dealt with, and (iii) and(i.c) follow. Case (i.d) (resp. (ii.b)) is similar : one considers almost elements and replacesE∗ by a freeA∗-module (resp. andF∗ by a finitely generatedA∗-module). In case (ii.a) (resp.(i.a)), for every finitely generated submodulem0 of m we can find, by proposition 2.3.11, afinitely generated (resp. presented)A-moduleF0 and a morphismF0 → F whose kernel andcokernel are annihilated bym0. It follows easily that we can replaceF by F0 and suppose thatF is finitely generated (resp. presented). Then the argument in [11, Ch.I§2 Prop.10] can betaken oververbatimto show (ii.a) (resp. (i.a)).

Lemma 2.4.26.LetB be anA-algebra.

(i) Let P be anA-module. If eitherP or B is almost finitely generated projective as anA-module, the natural morphism

B ⊗A alHomA(P,N)→ alHomB(B ⊗A P,B ⊗A N)(2.4.27)

is an isomorphism for allA-modulesN .(ii) Every almost finitely generated projectiveA-module is reflexive.(iii) If P is an almost finitely generated projectiveB-module, the natural morphism

alHomB(P,B)⊗B alHomA(B,A)→ alHomA(P,A) : φ⊗ ψ 7→ ψ φ(2.4.28)

is an isomorphism ofB-modules.

Proof. (i) is an easy consequence of lemma 2.4.24(i.b). To prove (ii), we apply the natural trans-formation (2.4.19) to (2.4.25) : by diagram chase one sees that the kernel and cokernel of themorphismF → (F ∗)∗ are killed byε. (iii) is analogous : one applies the natural transformation(2.4.28) to (2.4.25).

Lemma 2.4.29.Let Mn ; φn : Mn → Mn+1 | n ∈ N be a direct system ofA-modules andsuppose there exist sequencesεn | n ∈ N andδn | n ∈ N of ideals ofV such that

(i) limn→∞

εn = V and∏∞

j=0 δj is a Cauchy product (see(2.3.2.3.5));

(ii) εn ·alExtiA(Mn, N) = δn ·alExt

iA(Coker(φn), N) = 0 for all A-modulesN , all i > 0 and

all n ∈ N;(iii) δn ·Ker(φn) = 0 for all n ∈ N.

Thencolimn∈N

Mn is an almost projectiveA-module.

Proof. LetM = colimn∈N

Mn. By remark 2.4.9(i) it suffices to show thatalExtiA(M,N) vanishes

for all i > 0 and allA-modulesN . The mapsφn define a mapφ : ⊕nMn → ⊕nMn such that

we have a short exact sequence0→ ⊕nMn1−φ−→ ⊕nMn −→ M → 0. Applying the long exact

alExt sequence one obtains a short exact sequence (cp. [50, 3.5.10])

0→ limn∈N

1alExti−1A (Mn, N)→ alExtiA(M,N)→ limn∈N

alExtiA(Mn, N)→ 0.

Then lemma 2.4.2(ii) implies thatalExtiA(M,N) ≃ 0 for all i > 1 and moreoveralExt1A(M,N)is isomorphic tolim

n∈N

1alHomA(Mn, N). Let

φ∗n : alHomA(Mn+1, N)→ alHomA(Mn, N) f 7→ f φn


be the transpose ofφn and writeφn as a compositionMnpn−→ Im(φn)

qn→ Mn+1, so that

φ∗n = q∗np∗n, the composition of the respective transposed morphims. Wehave monomorphisms

Coker(p∗n) → alHomA(Ker(φn), N)Coker(q∗n) → alExt1A(Coker(φn), N)

for all n ∈ N. Henceδ2n · Coker(φ∗n) = 0 for all n ∈ N. Since

∏∞n=0 δ

2n is a Cauchy product,

lemma 2.4.2(iii) shows thatlimn∈N

1alHomA(Mn, N) ≃ 0 and the assertion follows.

Proposition 2.4.30.Suppose thatm is a flatV -module. Then for anyV -algebraR the functorM 7→M! commutes with tensor products and takes flatRa-modules to flatR-modules.

Proof. Let M be a flatRa-module andN → N ′ an injective map ofR-modules. Denote byK the kernel of the induced mapM! ⊗R N → M! ⊗R N ′; we haveKa ≃ 0. We obtain anexact sequence0 → m ⊗V K → m ⊗V M! ⊗R N → m ⊗V M! ⊗R N

′. But one sees easilythat m ⊗V K = 0 andm ⊗V M! ≃ M!, which shows thatM! is a flatR-module. Similarly,let M,N be twoRa-modules. Then the natural mapM∗ ⊗R N∗ → (M ⊗Ra N)∗ is an almostisomorphism and the assertion follows from remark 2.1.4(i).

2.5. Almost homotopical algebra. The formalism of abelian tensor categories provides a min-imal framework wherein the rudiments of deformation theorycan be developed.

2.5.2.5.1. Let(C ,⊗, U) be an abelian tensor category; we assume henceforth that⊗ is aright exact functor. LetA be a givenC -monoid. Then, for any two-sided idealI of A, thequotientA/I in the underlying abelian categoryC has a uniqueC -monoid structure such thatA→ A/I is a morphism of monoids.A/I is unitary ifA is. If I is a two-sided ideal ofA suchthatI2 = 0, then, using the right exactness of⊗ one checks thatI has a natural structure of anA/I-bimodule, unitary whenA is.

Definition 2.5.2. A C -extensionof aC -monoidB by aB-bimoduleI is a short exact sequenceof objects ofC

X : 0→ I → Cp→ B → 0(2.5.3)

such thatC is aC -monoid,p is a morphism ofC -monoids,I is a square zero two-sided idealin C and theE/I-bimodule structure onI coincides with the given bimodule structure onI.TheC -extensions form a categoryExmonC . The morphisms are commutative diagrams withexact rows

X :

0 // I //

f

Ep //

g

B //

h

0

X ′ : 0 // I ′ // E ′p′ // B′ // 0

such thatg andh are morphisms ofC -monoids. We letExmonC (B, I) be the subcategory ofExmonC consisting of allC -extensions ofB by I, where the morphisms are all short exactsequences as above such thatf := 1I andh := 1B.

2.5.2.5.4. We have also the variant in which all theC -monoids in (2.5.3) are required to beunitary (resp. to be algebras) andI is a unitaryB-bimodule (resp. whose left and rightB-module actions coincide,i.e. are switched by composition with the “commutativity constraints”θB|I andθI|B, see (2.2.2.2.5)); we will callExunC (resp.ExalC ) the corresponding category.


2.5.2.5.5. For a morphismφ : C → B of C -monoids, and aC -extensionX inExmonC (B, I),we can pullbackX via φ to obtain an exact sequenceX ∗ φ with a morphismφ∗ : X ∗ φ→ X;one checks easily that there exists a unique structure ofC -extension onX ∗ φ such thatφ∗

is a morphism ofC -extension; thenX ∗ φ is an object inExmonC (C, I). Similarly, given aB-linear morphismψ : I → J , we can push outX and obtain a well defined objectψ ∗X inExmonC (B, J) with a morphismX → ψ ∗X of ExmonC . In particular, ifI1 andI2 are twoB-bimodules, the functorspi∗ (i = 1, 2) associated to the natural projectionspi : I1 ⊕ I2 → Iiestablish an equivalence of categories

ExmonC (B, I1 ⊕ I2)∼→ ExmonC (B, I1)× ExmonC (B, I2)(2.5.6)

whose essential inverse is given by(E1, E2) 7→ (E1 ⊕ E2) ∗ δ, whereδ : B → B ⊕ B is thediagonal morphism. A similar statement holds forExal andExun. These operations can beused to induce an abelian group structure on the setExmonC (B, I) of isomorphism classes ofobjects ofExmonC (B, I) as follows. For any two objectsX, Y of ExmonC (B, I) we canformX ⊕ Y which is an object ofExmonC (B ⊕B, I ⊕ I). Letα : I ⊕ I → I be the additionmorphism ofI. Then we setX+Y := α∗(X⊕Y )∗δ. One can check thatX+Y ≃ Y +X foranyX, Y and that the trivial splitC -extensionB⊕I is a neutral element for+. Moreover everyisomorphism class has an inverse−X. The functorsX 7→ X ∗ φ andX 7→ ψ ∗ X commutewith the operation thus defined, and induce group homomorphisms

∗φ : ExmonC (B, I)→ ExmonC (C, I)ψ∗ : ExmonC (B, I)→ ExmonC (B, J).

2.5.2.5.7. We will need the variantExalC (B, I) defined in the same way, starting from thecategoryExalC (B, I). For instance, ifA is an almost algebra (resp. a commutative ring), wecan consider the abelian tensor categoryC = A-Mod. In this case theC -extensions will becalled simplyA-extensions, and we will writeExalA rather thanExalC . In fact the commuta-tive unitary case will soon become prominent in our work, andthe more general setup is onlyrequired for technical reasons, in the proof of proposition2.5.13 below, which is the abstractversion of a well-known result on the lifting of idempotentsover nilpotent ring extensions.

2.5.2.5.8. LetA be aC -monoid. We form the biproductA† := U ⊕A in C . We denote byp1,p2 the associated projections fromA† ontoU and respectivelyA. Also, let i1, i2 be the naturalmonomorphisms fromU , resp.A toA†. A† is equipped with a unitary monoid structure

µ† := i2 µ (p2 ⊗ p2) + i2 ℓ−1A (p1 ⊗ p2) + i2 r

−1A (p2 ⊗ p1) + i1 u

−1 (p1 ⊗ p1)

whereℓA, rA are the natural isomorphisms provided by [16, Prop. 1.3] andu : U → U⊗U is asin loc. cit. §1. In terms of the ringA†∗ ≃ U∗ ⊕ A∗ this is the multiplication(u1, b1) · (u2, b2) :=(u1 · u2, b1 · b2 + b1 · u2 + u1 · b2). Theni2 is a morphism of monoids and one verifies thatthe “restriction of scalars” functori∗2 defines an equivalence from the categoryA†-Uni.Mod ofunitaryA†-modules to the categoryA-Mod of all A-modules; letj denote the inverse functor.A similar discussion applies to bimodules.

2.5.2.5.9. Similarly, we derive equivalences of categories

ExunC (A†, j(M))

∗i2 //ExmonC (A,M)

(−)†oo

for all A-bimodulesM .


2.5.2.5.10. Next we specialise toA := U : for a givenU-moduleM let eM := σM/U ℓM :M → M ; working out the definitions one finds that the condition that(M,σM/U ) is a modulestructure is equivalent toe2M = eM . Let U × U be the product ofU by itself in the categoryof C -monoids. There is an isomorphism of unitaryC -monoidsζ : U † → U × U given byζ := i1 p1+ i2 p1+ i2 p2. Another isomorphism isτ ζ , whereτ is the flipi1 p2+ i2 p1.Hence we get equivalences of categories

U-Modj //

U †-Uni.Mod(ζ−1)∗

//

i∗2

oo (U × U)-Uni.Mod.(τζ)∗

oo

The compositioni∗2 (ζ−1 τ ζ)∗ j defines a self-equivalence ofU-Mod which associates

to a givenU-moduleM the newU-moduleMflip whose underlying object inC isM and suchthat eMflip = 1M − eM . The same construction applies toU-bimodules and finally we getequivalences

ExmonC (U,M)∼→ ExmonC (U,M

flip) X 7→ Xflip(2.5.11)

for all U-bimodulesM . If X := (0→ M → Eπ→ U → 0) is an extension andXflip := (0→

Mflip → Eflip → U → 0), then one verifies that there is a natural isomorphismXflip → Xof complexes inC inducing−1M on M , the identity onU and carrying the multiplicationmorphism onEflip to

−µE + ℓ−1E (π ⊗ 1E) + r−1E (1E ⊗ π) : E ⊗ E → E.

In terms of the associated rings, this corresponds to replacing the given multiplication(x, y) 7→x · y of E∗ by the new operation(x, y) 7→ π∗(x) · y + π∗(y) · x− x · y.

Lemma 2.5.12. If M is aU-bimodule whose left and right actions coincide, then everyexten-sion ofU byM splits uniquely.

Proof. Using the idempotenteM we get aU-linear decompositionM ≃ M1 ⊕ M2 wherethe bimodule structure onM1 is given by the zero morphisms and the bimodule structure onM2 is given byℓ−1M andr−1M . We have to prove thatExmonC (U,M) is equivalent to a one-point category. By (2.5.6) we can assume thatM = M1 or M = M2. By (2.5.11) we haveExmonC (U,M2) ≃ ExmonC (U,M

flip2 ) and onMflip

2 the bimodule actions are the zero mor-phisms. So it is enough to considerM =M1. In this case, ifX := (0→M → E → U → 0) isany extension,µE : E ⊗E → E factors through a morphismU ⊗U → E and composing withu : U → U ⊗ U we get a right inverse ofE → U , which shows thatX is the split extension.Then it is easy to see thatX does not have any non-trivial automorphisms, which proves theassertion.

Proposition 2.5.13.LetX := (0→ I → Ap→ A′ → 0) be aC -extension.

(i) Suppose thate′ ∈ A′∗ is an idempotent element whose left action on theA′-bimoduleIcoincides with its right action. Then there exists a unique idempotente ∈ A∗ such thatp∗(e) = e′.

(ii) Especially, ifA′ is unitary andI is a unitaryA′-bimodule, then every extension ofA′ by Iis unitary.

Proof. (i) : the hypothesise′2 = e′ implies thate′ : U → A′ is a morphism of (non-unitary)C -monoids. We can then replaceX byX ∗e′ and thereby assume thatA′ = U , p : A→ U andI isa (non-unitary)U-bimodule and the right and left actions onI coincide. The assertion to proveis that1U lifts to a unique idempotente ∈ A∗. However, this follows easily from lemma 2.5.12.To show (ii), we observe that, by (i), the unit1A′ of A′∗ lifts uniquely to an idempotente ∈ A∗.We have to show thate is a unit forA∗. Let us show the left unit property. Viae : U → A


we can view the extensionX as an exact sequence of leftU-modules. We can then splitX asthe direct sumX1 ⊕ X2 whereX1 is a sequence of unitaryU-modules andX2 is a sequenceof U-modules with trivial actions. But by hypothesis, onI and onA theU-module structure isunitary, soX = X1 and this is the left unit property.

2.5.2.5.14. So much for the general nonsense; we now return to almost algebras. As alreadyannounced,from here on, we assume throughout thatm is a flatV -module. As an immediateconsequence of proposition 2.5.13 we get natural equivalences of categories

ExalA1(B1,M1)×ExalA2(B2,M2)∼→ ExalA1×A2(B1 × B2,M1 ⊕M2)(2.5.15)

wheneverA1,A2 areV a-algebras,Bi is aAi-algebra andMi is a (unitary)Bi-module,i = 1, 2.

2.5.2.5.16. Notice that, ifA := Ra for someV -algebraR, S (resp.J) is aR-algebra (resp.anS-module) andX is any object ofExalR(S, J), then by applying termwise the localisationfunctor we get an objectXa ofExalA(Sa, Ja). With this notation we have the following lemma.

Lemma 2.5.17.LetB be anyA-algebra andI aB-module.

(i) The natural functor

ExalA!!(B!!, I∗)→ ExalA(B, I) X 7→ Xa(2.5.18)

is an equivalence of categories.(ii) The equivalence(2.5.18)induces a group isomorphismExalA!!

(B!!, I∗)∼→ ExalA(B, I)

functorial in all arguments.

Proof. Of course (ii) is an immediate consequence of (i). To show (i), let X := (0 → I →E → B → 0) be any object ofExalA(B, I). Using corollary 2.2.22 one sees easily that thesequenceX! := (0→ I! → E!! → B!! → 0) is right exact;X! won’t be exact in general, unlessB (and thereforeE) is an exact algebra. In any case, the kernel ofI! → E!! is almost zero, sowe get an extension ofB!! by a quotient ofI! which maps toI∗. In particular we get by pushoutan extensionX!∗ by I∗, i.e. an object ofExalA!!

(B!!, I∗) and in fact the assignmentX 7→ X!∗ isan essential inverse for the functor (2.5.18).

Remark 2.5.19. By inspecting the proof, we see that one can replaceI∗ by I!∗ := Im(I! → I∗)in (i) and (ii) above. WhenB is exact, alsoI! will do.

In [34, II.1.2] it is shown how to associate to any ring homomorphismR → S a naturalsimplicial complex ofS-modules denotedLS/R and called the cotangent complex ofS overR.

Definition 2.5.20. Let A → B be a morphism of almostV -algebras. Thealmost cotangentcomplexof B overA is the simplicialB!!-module

LB/A := B!! ⊗(V a×B)!! L(V a×B)!!/(V a×A)!! .

2.5.2.5.17. Usually we will want to viewLB/A as an object of the derived categoryD•(s.B!!)of simplicialB!!-modules. Indeed, the hyperext functors computed in this category relate thecotangent complex to a number of important invariants. Recall that, for any simplicial ringRand any twoR-modulesE, F the hyperext ofE andF is the abelian group defined as

ExtpR(E, F ) := colimn≥−p

HomD•(R)(σnE, σn+pF )

(whereσ is the suspension functor of [34, I.3.2.1.4]).Let us fix an almost algebraA. First we want to establish the relationship with differentials.

Definition 2.5.21. LetB be anyA-algebra,M anyB-module.


(i) An A-derivationof B with values inM is anA-linear morphism∂ : B → M such that∂(b1 · b2) = b1 · ∂(b2)+ b2 · ∂(b1) for b1, b2 ∈ B∗. The set of allM-valuedA-derivations ofB forms aV -moduleDerA(B,M) and the almostV -moduleDerA(B,M)a has a naturalstructure ofB-module.

(ii) We reserve the notationIB/A for the idealKer(µB/A : B ⊗A B → B). Themodule ofrelative differentialsof φ is defined as the (left)B-moduleΩB/A := IB/A/I

2B/A. It is

endowed with a naturalA-derivationδ : B → ΩB/A defined byb 7→ 1 ⊗ b− b⊗ 1 for allb ∈ B∗. The assignment(A→ B) 7→ ΩB/A defines a functor

Ω : V a-Alg.Morph→ V a-Alg.Mod

from the category of morphismsA → B of almostV -algebras to the category denotedV a-Alg.Mod, consisting of all pairs(B,M) whereB is an almostV -algebra andMis aB-module. The morphisms inV a-Alg.Morph are the commutative squares; themorphisms(B,M)→ (B′,M ′) in V a-Alg.Mod are all pairs(φ, f) whereφ : B → B′ isa morphism of almostV -algebras andf : B′ ⊗B M → M ′ is a morphism ofB′-modules.

2.5.2.5.22. The module of relative differentials enjoys the familiar universal properties thatone expects. In particularΩB/A represents the functorDerA(B,−), i.e. for any (left)B-moduleM the morphism

HomB(ΩB/A,M)→ DerA(B,M) f 7→ f δ(2.5.23)

is an isomorphism. As an exercise, the reader can supply the proof for this claim and for thefollowing standard proposition.

Proposition 2.5.24.LetB andC be twoA-algebras.(i) There is a natural isomorphism:

ΩC⊗AB/C ≃ C ⊗A ΩB/A.

(ii) Suppose thatC is aB-algebra. Then there is a natural exact sequence ofC-modules:

C ⊗B ΩB/A → ΩC/A → ΩC/B → 0.

(iii) LetI be an ideal ofB and letC := B/I be the quotientA-algebra. Then there is a naturalexact sequence:I/I2 → C ⊗B ΩB/A → ΩC/A → 0.

(iv) The functorΩ : V a-Alg.Morph→ V a-Alg.Mod commutes with all colimits.

We supplement these generalities with one more statement which is in the same vein aslemma 2.3.20 and which will be useful in section 5.3 to calculate the Fitting ideals of modulesof differentials.

Lemma 2.5.25.Let φ : B → B′ be a morphism ofA-algebras such thatI · Ker(φ) = I ·Coker(φ) = 0 for an idealI ⊂ A. Let dφ : ΩB/A ⊗B B

′ → ΩB′/A be the natural morphism.ThenI · Coker(dφ) = 0 andI4 ·Ker(dφ) = 0.

Proof. We will use the standard presentation

H(B/A) : B ⊗A B ⊗A B∂→ B ⊗A B

d→ ΩB/A → 0(2.5.26)

whered is defined by :b1 ⊗ b2 7→ b1 · db2 and∂ is the differential of the Hochschild complex :

b1 ⊗ b2 ⊗ b3 7→ b1b2 ⊗ b3 − b1 ⊗ b2b3 + b1b3 ⊗ b2.

By naturality ofH(B), we deduce a morphism of complexes :B′ ⊗B H(B/A) → H(B′/A).Then, by snake lemma, we derive an exact sequence :Ker(1B′ ⊗A φ) → Ker(dφ) → X,whereX is a quotient ofB′ ⊗A Coker(φ ⊗A φ). Using the Tor exact sequences we see thatKer(1B′ ⊗A φ) is annihilated byI2. It follows easily thatI4 annihilatesKer(dφ). Similarly,Coker(dφ) is a quotient ofCoker(1B′ ⊗A φ), soI · Coker(dφ) = 0.


Lemma 2.5.27.For anyA-algebraB there is a natural isomorphism ofB!!-modules

(ΩB/A)! ≃ ΩB!!/A!!.

Proof. Using the adjunction (2.5.23) we are reduced to showing thatthe natural map

φM : DerA!!(B!!,M)→ DerA(B,M

a)

is a bijection for allB!!-modulesM . Given∂ : B →Ma we construct∂! : B! →Ma! →M . We

extend∂! to V ⊕ B! by setting it equal to zero onV . Then it is easy to check that the resultingmap descends toB!!, hence giving anA-derivationB!! → M . This procedure yields a rightinverseψM to φM . To show thatφM is injective, suppose that∂ : B!! → M is an almost zeroA-derivation. Composing with the naturalA-linear mapB! → B!! we obtain an almost zeromap∂′ : B! → M . But m · B! = B!, hence∂′ = 0. This implies that in fact∂ = 0, and theassertion follows.

Proposition 2.5.28.LetM be aB-module. There exists a natural isomorphism ofB!!-modules

Ext0B!!(LB/A,M!) ≃ DerA(B,M).

Proof. To ease notation, setA := V a ×A andB := V a × B. We have natural isomorphisms :

Ext0B!!(LB/A,M!)≃ Ext0

B!!(LB!!/A!!

,M!) by [34, I.3.3.4.4]

≃ DerA!!(B!!,M!) by [34, II.1.2.4.2]

≃ DerA(B,M) by lemma 2.5.27.

But it is easy to see that the natural mapDerA(B,M)→ DerA(B,M) is an isomorphism.

Theorem 2.5.29.There is a natural isomorphism

ExalA(B,M)∼→ Ext1B!!

(LB/A,M!).

Proof. With the notation of the proof of proposition 2.5.28 we have natural isomorphisms

Ext1B!!(LB/A,M!)≃ Ext1

B!!(LB!!/A!!

,M!) by [34, I.3.3.4.4]

≃ ExalA!!(B!!,M!) by [34, III.1.2.3]

≃ ExalA(B,M)

where the last isomorphism follows directly from lemma 2.5.17(ii) and the subsequent remark2.5.19. Finally, (2.5.15) shows thatExalA(B,M) ≃ ExalA(B,M), as required.

Moreover we have the following transitivity theorem as in [34, II.2.1.2].

Theorem 2.5.30.LetA → B → C be a sequence of morphisms of almostV -algebras. Thereexists a natural distinguished triangle ofD•(s.C!!)

C!! ⊗B!!LB/A

u→ LC/A

v→ LC/B → C!! ⊗B!!

σLB/A

where the morphismsu andv are obtained by functoriality ofL.

Proof. It follows directly from loc. cit.

Proposition 2.5.31.Let (Aλ → Bλ)λ∈I be a system of almostV -algebra morphisms indexedby a small filtered categoryI. Then there is a natural isomorphism inD•(s.colim

λ∈IBλ!!)

colimλ∈I

LBλ/Aλ≃ Lcolim

λ∈IBλ/colim

λ∈IAλ.

Proof. Remark 2.2.28 gives an isomorphism :colimλ∈I

Aλ!!∼→ (colim

λ∈IAλ)!! (and likewise for

colimλ∈I

Bλ). Then the claim follows from [34, II.1.2.3.4].


Next we want to prove the almost version of the flat base changetheorem [34, II.2.2.1]. Tothis purpose we need some preparation.

Proposition 2.5.32.LetB andC be twoA-algebras and setTi := TorA!!i (B!!, C!!). If A, B,

C andB ⊗A C are all exact, then for everyi > 0 the natural morphismm ⊗V Ti → Ti is anisomorphism.

Proof. For any almostV -algebraD we letkD denote the complex ofD!!-modules[m⊗V D!! →D!!] placed in degrees−1, 0; we have a distiguished triangle

T (D) : m⊗V D!! → D!! → kD → m⊗V D!![1].

By the assumption, the natural mapkA → kB is a quasi-isomorphism andm ⊗V B!! ≃ B!. Onthe other hand, for alli ∈ N we have

TorA!!i (kB, C!!) ≃ TorA!!

i (kA, C!!) ≃ H−i(kA ⊗A!!C!!) = H−i(kC).

In particularTorA!!i (kB, C!!) = 0 for all i > 1. As m is flat overV , we havem ⊗V Ti ≃

TorA!!i (m ⊗V B!!, C!!). Then by the long exact Tor sequence associated toT (B)

L

⊗A!!C!! we

get the assertion for alli > 1. Next we consider the natural map of distinguished triangles

T (A)L

⊗A!!A!! → T (B)

L

⊗A!!C!!; writing down the associated morphism of long exact Tor

sequences, we obtain a diagram with exact rows :

0 // TorA!!1 (kA, A!!)

∂ //

(m⊗V A!!)⊗A!!A!!

i //

A!! ⊗A!!A!!

TorA!!1 (kB, C!!)

∂′ // (m⊗V B!!)⊗A!!C!!

i′ // B!! ⊗A!!C!!.

By the above, the leftmost vertical map is an isomorphism; moreover, the assumption givesKer(i) ≃ Ker(m → V ) ≃ Ker(i′). Then, since∂ is injective, also∂′ must be injective, whichimplies our assertion for the remaining casei = 1.

Corollary 2.5.33. Keep the notation of proposition 2.5.32 and suppose thatTorAi (B,C) ≃ 0for somei > 0. Then the correspondingTi vanishes.

Theorem 2.5.34.LetB,A′ be twoA-algebras. Suppose that the natural morphismBL

⊗AA′ →B′ := B ⊗A A

′ is an isomorphism inD•(s.A). Then the natural morphisms

B′!! ⊗B!!LB/A → LB′/A′

(B′!! ⊗B!!LB/A)⊕ (B′!! ⊗A′

!!LA′/A)→ LB′/A

are quasi-isomorphisms.

Proof. Let us remark that the functorD 7→ V a×D : A-Alg→ (V a×A)-Alg commutes withtensor products; hence the same holds for the functorD 7→ (V a × D)!! (see remark 2.2.28).Then, in view of corollary 2.5.33, the theorem is reduced immediately to [34, II.2.2.1].

As an application we obtain the vanishing of the almost cotangent complex for a certain classof morphisms.

Theorem 2.5.35.LetR→ S be a morphism of almost algebras such that

TorRi (S, S) ≃ 0 ≃ TorS⊗RSi (S, S) for all i > 0

(for the naturalS ⊗R S-module structure induced byµS/R). ThenLS/R ≃ 0 in D•(S!!).


Proof. SinceTorRi (S, S) = 0 for all i > 0, theorem 2.5.34 applies (withA := R andB :=A′ := S), giving the natural isomorphisms

(S ⊗R S)!! ⊗S!!LS/R ≃ LS⊗RS/S

((S ⊗R S)!! ⊗S!!LS/R)⊕ ((S ⊗R S)!! ⊗S!!

LS/R) ≃ LS⊗RS/R.(2.5.36)

SinceTorS⊗RSi (S, S) = 0, the same theorem also applies withA := S ⊗R S,B := S, A′ := S,

and we notice that in this caseB′ ≃ S; hence we have

LS/S⊗RS ≃ S!! ⊗S!!LS/S⊗RS ≃ LS/S ≃ 0.(2.5.37)

Next we apply transitivity to the sequenceR→ S ⊗R S → S, to obtain (thanks to (2.5.37))

S!! ⊗S⊗RS!!LS⊗RS/R ≃ LS/R.(2.5.38)

ApplyingS!! ⊗S⊗RS!!− to the second isomorphism (2.5.36) we obtain

LS/R ⊕ LS/R ≃ S!! ⊗S⊗RS!!LS⊗RS/R.(2.5.39)

Finally, composing (2.5.38) and (2.5.39) we derive

LS/R ⊕ LS/R∼→ LS/R.(2.5.40)

However, by inspection, the isomorphism (2.5.40) is the summap. ConsequentlyLS/R ≃ 0, asclaimed.

Finally we have a fundamental spectral sequence as in [34, III.3.3.2].

Theorem 2.5.41.Let φ : A → B be a morphism of almost algebras such thatB ⊗A B ≃ B(e.g.such thatB is a quotient ofA). Then there is a first quadrant homology spectral sequenceof bigraded almost algebras

E2pq := Hp+q(Symq

B(LaB/A))⇒ TorAp+q(B,B).

Proof. We replaceφ by 1V a × φ and apply the functorB 7→ B!! (which commutes with tensorproducts by remark 2.2.28) thereby reducing the assertion to [34, III.3.3.2].


3. ALMOST RING THEORY

3.1. Flat, unramified and etale morphisms. LetA→ B be a morphism of almostV -algebras.Using the natural “multiplication” morphism ofA-algebrasµB/A : B ⊗A B → B we can viewB as aB ⊗A B-algebra.

Definition 3.1.1. Let φ : A→ B be a morphism of almostV -algebras.

(i) We say thatφ is aflat (resp.faithfully flat, resp.almost projective) morphismif B is a flat(resp. faithfully flat, resp. almost projective)A-module.

(ii) We say thatφ is (uniformly) almost finite(resp.finite) if B is a (uniformly) almost finitelygenerated (resp. finitely generated)A-module.

(iii) We say thatφ isweakly unramified(resp.unramified) if B is a flat (resp. almost projective)B ⊗A B-module (via the morphismµB/A defined above).

(iv) φ is weaklyetale(resp.etale) if it is flat and weakly unramified (resp. unramified).

Furthermore, in analogy with definition 2.4.4, we shall write “(uniformly) almost finite projec-tive” to denote a morphismφ which is both (uniformly) almost finite and almost projective.

Lemma 3.1.2. Letφ : A→ B andψ : B → C be morphisms of almostV -algebras.

(i) LetA → A′ be any morphism ofV a-algebras; ifφ is flat (resp. almost projective, resp.faithfully flat, resp. almost finite, resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale) then the same holds forφ⊗A 1A′.

(ii) If both φ andψ are flat (resp. almost projective, resp. faithfully flat, resp. almost finite,resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale), then so isψφ.

(iii) If φ is flat andψ φ is faithfully flat, thenφ is faithfully flat.(iv) If φ is weakly unramified andψ φ is flat (resp. weaklyetale), thenψ is flat (resp. weakly

etale).(v) If φ is unramified andψ φ is etale, thenψ is etale.(vi) φ is faithfully flat if and only if it is a monomorphism andB/A is a flatA-module.

(vii) If φ is almost finite and weakly unramified, thenφ is unramified.

Proof. For (vi) use the Tor sequences. In view of proposition 2.4.13(ii), to show (vii) it sufficesto know thatB is an almost finitely presentedB ⊗A B-module; but this follows from theexistence of an epimorphism ofB ⊗A B-modules(B ⊗A B) ⊗A B → Ker(µB/A) defined byx ⊗ b 7→ x · (1 ⊗ b − b ⊗ 1). Of the remaining assertions, only (iv) and (v) are not obvious,but the proof is just the “almost version” of a well-known argument. Let us show (v); the sameargument applies to (iv). We remark thatµB/A is an etale morphism, sinceφ is unramified.DefineΓψ := 1C ⊗B µB/A. By (i), Γψ is etale. Define alsop := (ψ φ)⊗A 1B. By (i), p is flat(resp. etale). The claim follows by remarking thatψ = Γψ p and applying (ii).

Remark 3.1.3. (i) Suppose we work in the classical limit case, that is,m := V (cp. example2.1.2(ii)). Then we caution the reader that our notion of “etale morphism” is more generalthan the usual one, as defined in [29]. The relationship between the usual notion and ours isdiscussed in the digression (3.3.3.3.38).

(ii) The naive hope that the functorA 7→ A!! might preserve flatness is crushed by the fol-lowing counterexample. Let(V,m) be as in example 2.1.2(i) and letk be the residue fieldof V . Consider the flat mapV × V → V defined as(x, y) 7→ x. We get a flat morphismV a × V a → V a in V a-Alg; applying the left adjoint to localisation yields a mapV ×k V → Vthat is not flat. On the other hand, faithful flatnessis preserved. Indeed, letφ : A→ B be a mor-phism of almost algebras. Thenφ is a monomorphism if and only ifφ!! is injective; moreover,B!!/Im(A!!) ≃ B!/A!, which is flat overA!! if and only if B/A is flat overA, by proposition2.4.30.


We will find useful to study certain “almost idempotents”, asin the following proposition.

Proposition 3.1.4. A morphismφ : A → B is unramified if and only if there exists an almostelementeB/A ∈ B ⊗A B∗ such that

(i) e2B/A = eB/A;(ii) µB/A(eB/A) = 1;(iii) x · eB/A = 0 for all x ∈ IB/A∗.

Proof. Suppose thatφ is unramified. We start by showing that for everyε ∈ m there existalmost elementseε of B ⊗A B such that

e2ε = ε · eε µB/A(eε) = ε · 1 IB/A∗ · eε = 0.(3.1.5)

SinceB is an almost projectiveB ⊗A B-module, for everyε ∈ m there exists an “approximatesplitting” for the epimorphismµB/A : B ⊗A B → B, i.e. a B ⊗A B-linear morphismuε :B → B ⊗A B such thatµB/A uε = ε · 1B. Seteε := uε 1 : A → B ⊗A B. We see thatµB/A(eε) = ε · 1. To show thate2ε = ε · eε we use theB ⊗A B-linearity ofuε to compute

e2ε = eε · uε(1) = uε(µB/A(eε) · 1) = uε(µB/A(eε)) = ε · eε.

Next take any almost elementx of IB/A and compute

x · eε = x · uε(1) = uε(µB/A(x) · 1) = 0.

This establishes (3.1.5). Next let us take any otherδ ∈ m and a corresponding almost elementeδ.Bothε·1−eε andδ·1−eδ are elements ofIB/A∗, hence we have(δ·1−eδ)·eε = 0 = (ε·1−eε)·eδwhich implies

δ · eε = ε · eδ for all ε, δ ∈ m.(3.1.6)

Let us define a mapeB/A : m⊗V m→ B ⊗A B∗ by the rule

ε⊗ δ 7→ δ · eε for all ε, δ ∈ m.(3.1.7)

To show that (3.1.7) does indeed determine a well defined morphism, we need to check thatδ · v · eε = δ · ev·ε andδ · eε+ε′ = δ · (eε + eε′) for all ε, ε′, δ ∈ m and allv ∈ V . However, bothidentities follow easily by a repeated application of (3.1.6). It is easy to see thateB/A defines analmost element with the required properties.

Conversely, suppose an almost elementeB/A of B ⊗A B is given with the stated properties.We defineu : B → B ⊗A B by b 7→ eB/A · (1⊗ b) (b ∈ B∗) andv := µB/A. Then (iii) says thatu is aB ⊗A B-linear morphism and (ii) shows thatv u = 1B. Hence, by lemma 2.4.10,φ isunramified.

Remark 3.1.8. The proof of proposition 3.1.4 shows that ifI is an ideal in an almostV -algebraA, thenA/I is almost projective overA if and only if I is generated by an idempotent ofA∗.This idempotent is uniquely determined.

Corollary 3.1.9. Under the hypotheses and notation of the proposition, the ideal IB/A has anatural structure ofB ⊗A B-algebra, with unit morphism given by1 := 1B⊗AB/A

− eB/A andwhose multiplication is the restriction ofµB⊗AB/A to IB/A. Moreover the natural morphism

B ⊗A B → IB/A ⊕ B x 7→ (x · 1⊕ µB/A(x))

is an isomorphism ofB ⊗A B-algebras.

Proof. Left to the reader as an exercise.


3.2. Lifting theorems. Throughout the following, the terminology “epimorphism ofV a-alge-bras” will refer to a morphism ofV a-algebras that induces an epimorphism on the underlyingV a-modules.

Lemma 3.2.1. LetA → B be an epimorphism of almostV -algebras with kernelI. LetU betheA-extension0 → I/I2 → A/I2 → B → 0. Then the assignmentf 7→ f ∗ U defines anatural isomorphism

HomB(I/I2,M)

∼→ ExalA(B,M).(3.2.2)

Proof. LetX := (0 → M → Ep→ B → 0) be anyA-extension ofB byM . The composition

g : A→ Ep→ B of the structural morphism forE followed byp coincides with the projection

A→ B. Thereforeg(I) ⊂M andg(I2) = 0. Henceg factors throughA/I2; the restriction ofgto I/I2 defines a morphismf ∈ HomB(I/I

2,M) and a morphism ofA-extensionsf ∗U → X.In this way we obtain an inverse for (3.2.2).

3.2.3.2.3. Now consider any morphism ofA-extensions

B :

f

0 // I //

u

B //

f

B0//

f0

0

C : 0 // J // C // C0// 0.

(3.2.4)

The morphismu induces by adjunction a morphism ofC0-modules

C0 ⊗B0 I → J(3.2.5)

whose image is the idealI · C, so that the square diagram of almost algebras defined byf iscofibred (i.e.C0 ≃ C ⊗B B0) if and only if (3.2.5) is an epimorphism.

Lemma 3.2.6. Let f : B → C be a morphism ofA-extensions as above, such that the corre-sponding square diagram of almost algebras is cofibred. Thenthe morphismf : B → C is flatif and only iff0 : B0 → C0 is flat and(3.2.5)is an isomorphism.

Proof. It follows directly from the (almost version of the) local flatness criterion (see [41, Th.22.3]).

We are now ready to put together all the work done so far and begin the study of deformationsof almost algebras.

3.2.3.2.7. The morphismu : I → J is an element inHomB0(I, J); by lemma 3.2.1 the lattergroup is naturally isomorphic toExalB(B0, J). By applying transitivity (theorem 2.5.30) to the

sequence of morphismsB → B0f0→ C0 we obtain an exact sequence of abelian groups

ExalB0(C0, J)→ ExalB(C0, J)→ HomB0(I, J)∂→ Ext2C0!!

(LC0/B0, J!).

Hence we can form the elementω(B, f0, u) := ∂(u) ∈ Ext2C0!!(LC0/B0

, J!). The proof of thenext result goes exactly as in [34, III.2.1.2.3].

Proposition 3.2.8. Let theA-extensionB, theB0-linear morphismu : I → J and the mor-phism ofA-algebrasf0 : B0 → C0 be given as above.

(i) There exists anA-extensionC and a morphismf : B → C completing diagram(3.2.4)ifand only ifω(B, f0, u) = 0. (i.e. ω(B, f0, u) is the obstruction to the lifting ofB overf0.)

(ii) Assume that the obstructionω(B, f0, u) vanishes. Then the set of isomorphism classes ofA-extensionsC as in(i) forms a torsor under the group:

ExalB0(C0, J) ≃ Ext1C0!!(LC0/B0

, J!).


(iii) The group of automorphisms of anA-extensionC as in (i) is naturally isomorphic toDerB0(C0, J) (≃ Ext0C0!!

(LC0/B0 , J!)).

3.2.3.2.9. The obstructionω(B, f0, u) depends functorially onu. More exactly, if we denoteby

ω(B, f0) ∈ Ext2C0!!(LC0/B0 , (C0 ⊗B0 I)!)

the obstruction corresponding to the natural morphismI → C0 ⊗B0 I, then for any othermorphismu : I → J we have

ω(B, f0, u) = v! ω(B, f0)

wherev is the morphism (3.2.5). Taking lemma 3.2.6 into account we deduce

Corollary 3.2.10. Suppose thatB0 → C0 is flat. Then

(i) The classω(B, f0) is the obstruction to the existence of a flat deformation ofC0 overB,i.e. of aB-extensionC as in (3.2.4)such thatC is flat overB andC ⊗B B0 → C0 is anisomorphism.

(ii) If the obstructionω(B, f0) vanishes, then the set of isomorphism classes of flat deforma-tions ofC0 overB forms a torsor under the groupExalB0(C0, C0 ⊗B0 I).

(iii) The group of automorphisms of a given flat deformation ofC0 overB is naturally isomor-phic toDerB0(C0, C0 ⊗B0 I).

3.2.3.2.11. Now, suppose we are given twoA-extensionsC1, C2 with morphisms ofA-extensions

B :

f i

0 // I //

ui

B //

f i

B0//

f i0

0

C i : 0 // J i // C i // C i0

// 0

and morphismsv : J1 → J2, g0 : C10 → C2

0 such that

u2 = v u1 and f 20 = g0 f

10 .(3.2.12)

We consider the problem of finding a morphism ofA-extensions

C1 :

g

0 // J1 //

v

C1 //

g

C10

//

g0

0

C2 : 0 // J2 // C2 // C20

// 0

(3.2.13)

such thatf 2 = g f 1. Let us denote bye(C i) ∈ Ext1Ci0!!(LCi

0/B, J i! ) the classes defined by the

B-extensionsC1, C2 via the isomorphism of theorem 2.5.29 and by

v∗ : Ext1C10!!(LC1

0/B, J1

! )→ Ext1C10!!(LC1

0/B, J2

! )

∗g0 : Ext1C20!!(LC2

0/B, J2

! )→ Ext1C20!!(C2

0!! ⊗C10!!LC1

0/B, J2

! )

the canonical morphisms defined byv andg0. Using the natural isomorphism

Ext1C10!!(LC1

0/B, J2

! ) ≃ Ext1C20!!(C2

0!! ⊗C10!!LC1

0/B, J2

! )

we can identify the target of bothv∗ and∗g with Ext1C10!!(LC1

0/B, J2

! ). It is clear that the problem

admits a solution if and only if theA-extensionsv ∗ C1 and C2 ∗ g0 coincide, i.e. if andonly if v ∗ e(C1) − e(C2) ∗ g0 = 0. By applying transitivity to the sequence of morphismsB → B0 → C1

0 we obtain an exact sequence

Ext1C10!!(LC1

0/B0, J2

! ) → Ext1C10!!(LC1

0/B, J2

! )→ HomC10(C1

0 ⊗B0 I, J2)


It follows from (3.2.12) that the image ofv∗e(C1)−e(C2)∗g0 in the groupHomC10(C1

0⊗B0I, J2)

vanishes, therefore

v ∗ e(C1)− e(C2) ∗ g0 ∈ Ext1C10!!(LC1

0/B0, J2

! ).(3.2.14)

In conclusion, we derive the following result as in [34, III.2.2.2].

Proposition 3.2.15.With the above notations, the class(3.2.14)is the obstruction to the exis-tence of a morphism ofA-extensionsg : C1 → C2 as in(3.2.13)such thatf 2 = gf 1. When theobstruction vanishes, the set of such morphisms forms a torsor under the groupDerB0(C

10 , J

2)(the latter being identified withExt0C2

0!!(C2

0!! ⊗C10!!LC1

0/B0, J2

! )).

3.2.3.2.16. For a given almostV -algebraA, we define the categoryw.Et(A) (resp. Et(A))as the full subcategory ofA-Alg consisting of all weakly etale (resp. etale)A-algebras. Noticethat, by lemma 3.1.2(iv) all morphisms inw.Et(A) are weakly etale.

Theorem 3.2.17.LetA be aV a-algebra.

(i) LetB be a weaklyetaleA-algebra,C anyA-algebra andI ⊂ C a nilpotent ideal. Thenthe natural morphism

HomA-Alg(B,C)→ HomA-Alg(B,C/I)

is bijective.(ii) Let I ⊂ A a nilpotent ideal andA′ := A/I. Then the natural functor

w.Et(A)→ w.Et(A′) (φ : A→ B) 7→ (1A′ ⊗A φ : A′ → A′ ⊗A B)

is an equivalence of categories.(iii) The equivalence of (ii) restricts to an equivalenceEt(A)→ Et(A′).

Proof. By induction we can assumeI2 = 0. Then (i) follows directly from proposition 3.2.15and theorem 2.5.35. We show (ii) : by corollary 3.2.10 (and again theorem 2.5.35) a givenweakly etale morphismφ′ : A′ → B′ can be lifted to auniqueflat morphismφ : A → B.We need to prove thatφ is weakly etale,i.e. thatB is B ⊗A B-flat. However, it is clear thatµB′/A′ : B′ ⊗A′ B′ → B′ is weakly etale, hence it has a flat liftingµ : B ⊗A B → C. Thenthe compositionA → B ⊗A B → C is flat and it is a lifting ofφ′. We deduce that there is anisomorphism ofA-algebrasα : B → C lifting 1B′ and moreover the morphismsb 7→ µ(b⊗ 1)andb 7→ µ(1 ⊗ b) coincide withα. Claim (ii) follows. To show (iii), suppose thatA′ → B′

is etale and letIB′/A′ denote as usual the kernel ofµB′/A′. By corollary 3.1.9 there is a naturalmorphism of almost algebrasB′ ⊗A′ B′ → IB′/A′ which is clearly etale. HenceIB′/A′ lifts toa weakly etaleB ⊗A B-algebraC, and the isomorphismB′ ⊗A′ B′ ≃ IB′/A′ ⊕ B′ lifts to anisomorphismB ⊗A B ≃ C ⊕ B of B ⊗A B-algebras. It follows thatB is an almost projectiveB ⊗A B-module,i.e.A→ B is etale, as claimed.

We conclude with some results on deformations of almost modules. These can be establishedindependently of the theory of the cotangent complex, alongthe lines of [34, IV.3.1.12].

3.2.3.2.18. We begin by recalling some notation fromloc. cit. Let R be a ring andJ ⊂ Ran ideal withJ2 = 0. SetR′ := R/J ; an extension ofR-modulesM := (0 → K →

Mp→ M ′ → 0) whereK andM ′ are killed byJ , defines a natural morphism ofR′-modules

u(M) : J ⊗R′ M ′ → K such thatu(M)(x ⊗m′) = xm for x ∈ J , m ∈ M andp(m) = m′.By the local flatness criterion ([41, Th. 22.3])M is flat overR if and only ifM ′ is flat overR′

andu(M) is an isomorphism. One can then show the following.

Proposition 3.2.19. (cp. [34, IV.3.1.5])


(i) GivenR′-modulesM ′ andK and a morphismu′ : J ⊗R′ M ′ → K there exists an ob-structionω(R, u′) ∈ Ext2R′(M ′, K) whose vanishing is necessary and sufficient for theexistence of an extension ofR-modulesM ofM ′ byK such thatu(M) = u′.

(ii) Whenω(R, u′) = 0, the set of isomorphism classes of such extensionsM forms a torsorunderExt1R′(M ′, K); the group of automorphisms of such an extension is isomorphic toHomR′(M ′, K).

Lemma 3.2.20.Let A → B be a finite morphism of almost algebras with nilpotent kernel.There existsm ≥ 0 such that the following holds. For everyA-linear morphismφ : M → N ,setφB := φ⊗A 1B :M ⊗A B → N ⊗A B; then :

(i) AnnA(Coker(φB))m ⊂ AnnA(Coker(φ)).

(ii) (AnnV (Ker(φB)) · AnnV (TorA1 (B,N)) · AnnV (Coker(φ)))m ⊂ AnnA(Ker(φ)).

If B = A/I for some nilpotent idealI, andIn = 0, then we can takem = n in (i) and(ii) .

Proof. Under the assumptions, we can find a finitely generatedA∗-moduleQ such thatm ·B∗ ⊂Q ⊂ B∗. By [30, 1.1.5], there exists a finite filtration0 = Jm ⊂ ... ⊂ J1 ⊂ J0 = A∗ such thateachJi/Ji+1 is a quotient of a direct sum of copies ofQ. This implies that, for everyA-moduleM , we have

AnnA(M ⊗A B)m ⊂ AnnA(M).(3.2.21)

(i) follows easily. Notice that ifB = A/I andIn = 0, then we can takem = n in (3.2.21). For

(ii) let C• := Cone(φ). We estimateH := H−1(C•L

⊗A B) in two ways. By the first spectralsequence of hyperhomology we have an exact sequenceTorA1 (N,B)→ H → Ker(φB). By thesecond spectral sequence for hyperhomology we have an exactsequenceTorA2 (Coker(φ), B)→Ker(φ)⊗AB → H. HenceKer(φ)⊗AB is annihilated by the product of the three annihilatorsin (ii) and the result follows by applying (3.2.21) withM := Ker(φ).

Lemma 3.2.22.Keep the assumptions of lemma 3.2.20 and letM be anA-module.

(i) If A→ B is an epimorphism,M is flat andMB := B ⊗AM is almost projective overB,thenM is almost projective overA.

(ii) If MB is an almost finitely generatedB-module thenM is an almost finitely generatedA-module.

(iii) If TorA1 (B,M) = 0 andMB is almost finitely presented overB, thenM is almost finitelypresented overA.

Proof. (i) : we have to show thatExt1A(M,N) is almost zero for everyA-moduleN . LetI := Ker(A → B); by assumptionI is nilpotent, so by the usual devissage we may assumethatI · N = 0. If χ ∈ Ext1A(M,N) is represented by an extension0 → N → Q → M → 0then after tensoring byB and using the flatness ofM we get an exact sequence ofB-modules0 → N → B ⊗A Q → MB → 0. Thusχ comes from an element ofExt1B(MB, N) which isalmost zero by assumption.

(ii) : for a given finitely generated subidealm0 ⊂ m, let N ⊂ MB be a finitely generatedB-submodule such thatm0 · MB ⊂ N . Since the induced mapM∗ ⊗A∗ B∗ → (MB)∗ isalmost surjective, we can find a finitely generatedA-submoduleN0 ⊂ M such thatm0 · N ⊂Im((N0)B → MB); by lemma 3.2.20(i) it follows thatm2n

0 · (M/N0) = 0 for somen ≥ 0depending only onB, whence the claim.

(iii) : Let m0 be as above. By (ii),M is almost finitely generated overA, so we can choose amorphismφ : Ar →M such thatm0 ·Coker(φ) = 0. ConsiderφB := φ⊗A1B : Br →MB. Byclaim 2.3.13, there is a finitely generated submoduleN of Ker(φB) containingm2

0 · Ker(φB).Notice thatKer(φ)⊗AB maps ontoKer(Br → Im(φ)⊗AB) andKer(Im(φ)⊗AB →MB) ≃


TorA1 (B,Coker(φ)) is annihilated bym0. Hencem0 · Ker(φB) is contained in the image ofKer(φ) and therefore we can lift a finite generating setx′1, ..., x

′n for m2

0 ·N to almost elementsx1, ..., xn of Ker(φ). If we quotientAr by the span of thesexi, we get a finitely presentedA-moduleF with a morphismφ : F → M such thatKer(φ ⊗A B) is annihilated bym4

0 andCoker(φ) is annihilated bym0. By lemma 3.2.20(ii) we derivem5m

0 · Ker(φ) = 0 for somem ≥ 0. Sincem0 is arbitrary, this proves the result.

Remark 3.2.23. (i) Inspecting the proof, one sees that parts (ii) and (iii) of lemma 3.2.22 holdwhenever (3.2.21) holds. For instance, ifA → B is any faithfully flat morphism, then (3.2.21)holds withm := 1.

(ii) Consequently, ifA → B is faithfully flat andM is anA-module such thatMB is flat(resp. almost finitely generated, resp. almost finitely presented) overB, thenM is flat (resp.almost finitely generated, resp. almost finitely presented)overA.

(iii) On the other hand, we do not know whether a general faithfully flat morphismA → Bdescends almost projectivity. However, using (ii) and proposition 2.4.13 we see that if theB-moduleMB is almost finitely generated projective, thenM has the same property.

(iv) Furthermore, ifB is faithfully flat and almost finitely presented as anA-module, thenA → B does descend almost projectivity, as can be easily deduced from lemma 2.4.26(i) andproposition 2.4.13(ii).

Theorem 3.2.24.Let I ⊂ A be a nilpotent ideal, and setA′ := A/I. Suppose thatm is a (flat)V -module of homological dimension≤ 1. LetP ′ be an almost projectiveA′-module.

(i) There is an almost projectiveA-moduleP withA′ ⊗A P ≃ P ′.(ii) If P ′ is almost finitely presented, thenP is almost finitely presented.

Proof. As usual we reduce toI2 = 0. Then proposition 3.2.19(i) applies withR := A∗,J := I∗, R′ := A∗/I∗,M ′ := P ′! , K := I∗ ⊗R′ P ′! andu′ := 1K . We obtain a classω(A∗, u′) ∈Ext2R′(P ′! , I∗ ⊗R′ P ′! ) which gives the obstruction to the existence of a flatA∗-moduleF liftingP ′! . SinceP ′! is almost projective, we know thatm · Ext2R′(P ′! , I∗ ⊗R′ P ′! ) = 0, which saysthat0 = ε · ω(A∗, u′) = ω(A∗, ε · u′) for all ε ∈ m. In other words, for everyε ∈ m we canfind an extension ofA∗-modulesPε of P ′! by I∗ ⊗R′ P ′! such thatu(Pε) = ε · 1I∗⊗R′P ′

!. Let

χε ∈ Ext1A∗(P ′! , I∗ ⊗R′ P ′! ) be the class ofPε. Notice that, for anyδ ∈ m, δ · χε is the class of

an extensionX such thatu(X) = δ · u(Pε) = δ · ε · 1I∗⊗R′P ′!, hence, by proposition 3.2.19(ii),

γ · (δ · χε − χδ·ε) = 0 for all γ ∈ m. Hence we can define a morphism

χ : m⊗V m⊗V m→ Ext1A∗(P ′! , I∗ ⊗R′ P ′! ) ε⊗ δ ⊗ γ 7→ δ · γ · χε.

However, one sees easily thatm⊗V m⊗V m ≃ m andm⊗V P ′! ≃ P ′! , hence we can viewχ asan element ofHomV (m,Ext

1A∗(P ′! , I∗ ⊗R′ P ′! )) and moreover we have a spectral sequence

Epq2 := ExtpV (m,Ext

qA∗(P ′! , I∗ ⊗R′ P ′! ))⇒ Extp+qA∗

(P ′! , I∗ ⊗R′ P ′! )

with Epq2 = 0 for all p ≥ 2 (this spectral sequence is constructede.g. from the double complex

HomV (Fp,HomA∗(F′q, I∗⊗R′P ′! ))whereF• (resp.F ′•) is a projective resolution ofm (resp.P ′! )).

In particular, ourχ is an element inE012 which therefore survives in the abutment as a class of

E01∞ . The latter can be lifted to an elementχ via the surjectionExt1A∗

(P ′! , I∗⊗R′ P ′! )→ E01∞ . Let

0 → I∗ ⊗R′ P ′! → Q → P ′! → 0 be an extension representingχ. Checking compatibilities, wesee thatδ · ε · χ = δ ·χε for everyε, δ ∈ m. Henceu(χ) : I∗⊗R′ P ′! → I∗⊗R′ P ′! coincides withthe identity map on the submodulem · I∗⊗R′ P ′! . Sincem ·P! = P!, we see thatu(χ) is actuallythe identity map. By the local flatness criterion, it then follows thatQ is flat overR, hence theA-moduleP = Qa is a flat lifting ofP ′, so it is almost projective, by lemma 3.2.22(i). Now (ii)follows from (i), lemma 3.2.22(ii) and proposition 2.4.13(i).


Remark 3.2.25. (i) According to proposition 2.1.12(ii), theorem 3.2.24 applies especially whenm is countably generated as aV -module.

(ii) For P andP ′ as in theorem 3.2.24(ii) letσP : P → P ′ be the projection. It is natural to askwhether the pair(P, σP ) is uniquely determined up to isomorphism,i.e. whether, for any otherpair (Q, σQ : Q → P ′) for which theorem 3.2.24 holds, there exists anA-linear isomorphismφ : P → Q such thatσQ φ = σP . The answer is negative in general. Consider the caseP ′ := A′. TakeP := Q := A and letσP be the natural projection, whileσQ := (u′ · 1A′) σP ,whereu′ is a unit inA′∗. Then the uniqueness question amounts to whether every unitin A′∗lifts to a unit ofA∗. The following counterexample is related to the fact that the completion ofthe algebraic closureQp of Qp is not maximally complete. LetV := Zp, the integral closure ofZp in Qp. ThenV is a non-discrete valuation ring of rank one, and we take form the maximalideal ofV , A := (V/p2V )a andA′ := A/pA. Choose a compatible system of roots ofp. Analmost element ofA′ is just aV -linear morphismφ : colim

n>0p1/n!V → V/pV . Such aφ can be

represented (in a non-unique way) by an infinite series of theform∑∞

n=1 anp1−1/n! (an ∈ V ).

The meaning of this expression is as follows. For everym > 0, scalar multiplication by theelement

∑mn=1 anp

1−1/n! ∈ V defines a morphismφm : p1/m!V → V/pV . Form′ > m, letjm,m′ : p1/m!V → p1/m

′!V be the imbedding. Then we haveφm′ jm,m′ = φm, so that we candefineφ := colim

m>0φm. Similarly, every almost element ofA can be represented by an expression

of the forma0+∑∞

n=1 anp2−1/n!. Now, if σ : A→ A′ is the natural projection, the induced map

σ∗ : A∗ → A′∗ is given by:a0 +∑∞

n=1 anp2−1/n! 7→ a0. In particular, its image is the subring

V/p ⊂ (V/p)∗ = A′∗. For instance, the unit∑∞

n=1 p1−1/n! of A′∗ does not lie in the image of this

map.

In the light of the above remark, the best one can achieve in general is the following result.

Proposition 3.2.26.Assume (A) (see(2.1.2.1.6)) and keep the notation of theorem 3.2.24. Sup-pose that(Q, σQ : Q→ P ′) and(P, σP : P → P ′) are two pairs as in remark 3.2.25. Then foreveryε ∈ m there existA-linear morphismstε : P → Q andsε : Q→ P such that

PQ(ε)σQ tε = ε · σP σP sε = ε · σQsε tε = ε2 · 1P tε sε = ε2 · 1Q.

Proof. Since bothQ andP are almost projective andσP , σQ are epimorphisms, there existmorphismstε : P → Q andsε : Q → P such thatσQ tε = ε · σP andσP sε = ε · σQ.Then we haveσP (sε tε − ε2 · 1P ) = 0 andσQ (tε sε − ε2 1Q) = 0, i.e. the morphismuε := ε2 ·1P −sε tε (resp.vε := ε2 ·1Q− tε sε) has image contained in the almost submoduleIP (resp.IQ). SinceIm = 0 this impliesumε = 0 andvmε = 0. Hence

ε2m · 1P = (ε21P )m − umε = (

m−1∑

a=0

ε2aum−1−aε ) sε tε.

Defines(2m−1)ε := (∑m−1

a=0 ε2aum−1−aε ) sε. Notice thats(2m−1)ε = sε (

∑m−1a=0 ε

2avm−1−aε ).This implies the equalitiess(2m−1)ε tε = ε2m · 1P andtε s(2m−1)ε = ε2m · 1Q. Then the pair(s(2m−1)ε, ε

2(m−1) · tε) satisfiesPQ(ε2m−1). Under (A), every element ofm is a multiple of anelement of the formε2m−1, therefore the claim follows for arbitraryε ∈ m.

3.3. Descent. Faithfully flat descent in the almost setting presents no particular surprises: sincethe functorA 7→ A!! preserves faithful flatness of morphisms (see remark 3.1.3)many well-known results for usual rings and modules extendverbatimto almost algebras.

3.3.3.3.1. So for instance, faithfully flat morphisms are ofuniversal effective descent for thefibred categoriesF : V a-Alg.Modo → V a-Algo andG : V a-Alg.Morpho → V a-Algo


(see definition 2.5.21: for an almostV -algebraB, the fibreFB (resp.GB) is the opposite ofthe category ofB-modules (resp.B-algebras)). Then, using remark 3.2.23, we deduce alsouniversal effective descent for the fibred subcategories offlat (resp. almost finitely generated,resp. almost finitely presented, resp. almost finitely generated projective) modules. Likewise,a faithfully flat morphism is of universal effective descentfor the fibred subcategoriesEt

o→

V a-Algo of etale (resp.w.Eto→ V a-Algo of weakly etale) algebras.

3.3.3.3.2. More generally, since the functorA 7→ A!! preserves pure morphisms in the senseof [44], and since, by a theorem of Olivier (loc. cit.), pure morphisms are of universal effectivedescent for modules, the same holds for pure morphisms of almost algebras.

3.3.3.3.3. Non-flat descent is more delicate. Our results are not as complete here as it couldbe wished, but nevertheless, they suffice for current applications (namely, for the cases neededin [24]). Our first statement is the almost version of a theorem of Gruson and Raynaud (cp. [31,Part II, Th. 1.2.4]).

Proposition 3.3.4. A finite monomorphism of almost algebras descends flatness.

Proof. Let φ : A → B be such a morphism. Under the assumption, we can find a finiteA∗-moduleQ such thatm · B∗ ⊂ Q ⊂ B∗. One sees easily thatQ is a faithfulA∗-module, so by[31, Part II, Th. 1.2.4 and lemma 1.2.2],Q satisfies the following condition :

If (0 → N → L → P → 0) is an exact sequence ofA∗-modules withL flat, suchthatIm(N ⊗A∗ Q) is a pure submodule ofL⊗A∗ Q, thenP is flat.(3.3.5)

Now letM be anA-module such thatM ⊗AB is flat. Pick an epimorphismp : F → M with Ffree overA. ThenY := (0→ Ker(p⊗A 1B)→ F ⊗AB → M ⊗AB → 0) is universally exactoverB, hence overA. Consider the sequenceX := (0→ Im(Ker(p)! ⊗A∗ Q)→ F! ⊗A∗ Q→M!⊗A∗Q→ 0). ClearlyXa ≃ Y . However, it is easy to check that a sequenceE ofA-modulesis universally exact if and only if the sequenceE! is universally exact overA∗. We concludethatX = (Xa)! is a universally exact sequence ofA∗-modules, hence, by condition (3.3.5),M!

is flat overA∗, i.e.M is flat overA as required.

Corollary 3.3.6. LetA→ B be a finite morphism of almost algebras, with nilpotent kernel. IfC is a flatA-algebras such thatC ⊗A B is weaklyetale (resp.etale) overB, thenC is weaklyetale (resp.etale) overA.

Proof. In the weakly etale case, we have to show that the multiplication morphismµ : C ⊗AC → C is flat. AsN := Ker(A → B) is nilpotent, the local flatness criterion reduces thequestion to the situation overA/N . So we may assume thatA→ B is a monomorphism. ThenC⊗AC → (C⊗AC)⊗AB is a monomorphism, butµ⊗C⊗AC 1(C⊗AC)⊗AB is the multiplicationmorphism ofC ⊗A B, which is flat by assumption. Therefore, by proposition 3.3.4,µ is flat.

For the etale case, we have to show thatC is almost finitely presented as aC⊗AC-module. ByhypothesisC⊗AB is almost finitely presented as aC⊗AC⊗AB-module and we know alreadythatC is flat as aC ⊗A C-module, so by lemma 3.2.22(iii) (applied to the finite morphismC ⊗A C → C ⊗A C ⊗A B) the claim follows.

3.3.3.3.7. Next we consider the following situation. We aregiven a cartesian diagram ofalmost algebras

A0f2 //

f1

A2

g2

A1

g1 // A3

(3.3.8)


such that one of the morphismsAi → A3 (i = 1, 2) is an epimorphism. We denote byMi (resp.Mi,fl, resp. Mi,proj) the category of all (resp. flat, resp. almost projective)Ai-modules, fori = 0, ..., 3. Diagram (3.3.8) induces an essentially commutative diagram for the correspondingcategoriesMi, where the arrows are given by the “extension of scalars” functors. There followsa natural functor

π : M0 →M1 ×M3 M2

from M0 to the 2-fibred products ofM1 andM2 over M3. Recall (see [4, Ch.VII§3]) thatM1×M3M2 is the category whose objects are the triples(M1,M2, ξ), whereMi is anAi-module(i = 1, 2) andξ : A3⊗A1 M1

∼→ A3⊗A2 M2 is anA3-linear isomorphism. Given such an object

(M1,M2, ξ), let us denoteM3 := A3 ⊗A2 M2; we have a natural morphismM2 → M3, andξgives a morphismM1 →M3, so we can form the fibre productT (M1,M2, ξ) :=M1×M3M2. Inthis way we obtain a functorT : M1×M3 M2 →M0, and we leave to the reader the verificationthatT is right adjoint toπ. Let us denote byε : 1M0 → T π andη : π T → 1M1×M3

M2 theunit and counit of the adjunction.

Lemma 3.3.9. The functorπ induces an equivalence of full subcategories :

X ∈ Ob(M0) | εX is an isomorphismπ→ Y ∈ Ob(M1 ×M3 M2) | ηY is an isomorphism

havingT as essential inverse.

Proof. General nonsense.

Lemma 3.3.10.LetM be anyA0-module. ThenεM is an epimorphism. IfM is flat overA0,εM is an isomorphism.

Proof. Indeed,εM :M → (A1⊗A0 M)×A3⊗A0M (A2⊗A0 M) is the natural morphism. So, the

assertions follow by applying−⊗A0 M to the short exact sequence ofA0-modules

0→ A0f→ A1 ⊕A2

g→ A3 → 0(3.3.11)

wheref(a) := (f1(a), f2(a)) andg(a, b) := g1(a)− g2(b).

There is another case of interest, in whichεM is an isomorphism. Namely, suppose that oneof the morphismsAi → A3 (i = 1, 2), sayA1 → A3, has a section. Then also the morphismA0 → A2 gains a sections : A2 → A0 and we have the following :

Lemma 3.3.12. In the above situation, suppose that theA0-moduleM arises by extension ofscalars from anA2-moduleM ′, via the sections : A2 → A0. ThenεM is an isomorphism.

Proof. Indeed, in this case, (3.3.11) is split exact as a sequence ofA2-modules, and it remainssuch after tensoring byM ′.

Lemma 3.3.13.η(M1,M2,ξ) is an isomorphism for all objects(M1,M2, ξ).

Proof. To fix ideas, suppose thatA1 → A3 is an epimorphism. Consider any object(M1,M2, ξ)of M1 ×M3 M2. LetM := T (M1,M2, ξ); we deduce a natural morphism

φ : (M ⊗A0 A1)×M⊗A0A3 (M ⊗A0 A2)→ M1 ×M3 M2

such thatφ εM = 1M . It follows thatεM is injective, hence it is an isomorphism, by lemma3.3.10. We derive a commutative diagram with exact rows :

0 // M // (M ⊗A0 A1)⊕ (M ⊗A0 A2)

φ1⊕φ2

// M ⊗A0 A3//

φ3

0

0 // M // M1 ⊕M2// M3

// 0.


From the snake lemma we deduce

(∗) Ker(φ1)⊕Ker(φ2) ≃ Ker(φ3)(∗∗) Coker(φ1)⊕ Coker(φ2) ≃ Coker(φ3).

SinceM3 ≃M1⊗A1A3 we haveA3⊗A1Coker(φ1) ≃ Coker(φ3). But by assumptionA1 → A3

is an epimorphism, so alsoCoker(φ1) → Coker(φ3) is an epimorphism. Then(∗∗) impliesthat Coker(φ2) = 0. But φ3 = 1A3 ⊗A2 φ2, thusCoker(φ3) = 0 as well. We look at the

exact sequence0 → Ker(φ1) → M ⊗A0 A1φ1→ M1 → 0 : applyingA3 ⊗A1 − we obtain an

epimorphismA3⊗A1Ker(φ1)→ Ker(φ3). From(∗) it follows thatKer(φ2) = 0. In conclusion,φ2 is an isomorphism. Hence the same is true forφ3 = 1A3 ⊗A2 φ2, and again(∗), (∗∗) showthatφ1 is an isomorphism as well, which implies the claim.

Lemma 3.3.14. If (A1 × A2)⊗A0 M is flat overA1 × A2, thenM is flat overA0.

Proof. Suppose thatA1 → A3 is an epimorphism and letI be its kernel. LetA := A1!!×A3!!A2!!;

it suffices to show thatM! is a flat A-module. However, in view of proposition 2.4.30, theassumption implies that(A1!! × A2!!)⊗A M! is a flatA1!! × A2!!-module.I! is the kernel of theepimorphismA1!! → A3!!. Moreover,I! identifies naturally with an ideal ofA andA/I! ≃ A2!!.Then the desired conclusion follows from [25, lemma].

Proposition 3.3.15.The functorπ restricts to equivalences :

M0,fl∼→M1,fl ×M3,fl

M2,fl

M0,proj∼→M1,proj ×M3,proj

M2,proj.

Proof. The assertion for flat almost modules follows directly from lemmata 3.3.9, 3.3.10, 3.3.13and 3.3.14. SetB := A1 × A2. To establish the second equivalence, it suffices to show that, ifP is anA0-module such thatB ⊗A0 P is almost projective overB, thenP is almost projectiveoverA0, or which is the same, thatalExtiA0

(P,N) ≃ 0 for all i > 0 and anyA0-moduleN . We know already thatP is flat. LetM be anyA0-module andN anyB-module. The

standard isomorphismRHomB(BL

⊗A0M,N) ≃ RHomA0(M,N) yields a natural isomorphismalExtiB(B ⊗A0 M,N) ≃ alExtiA0

(M,N), wheneverTorA0j (B,M) = 0 for everyj > 0. In

particular, we havealExtiA0(P,N) ≃ 0 wheneverN comes from either anA1-module, or an

A2-module. For a generalA0-moduleN there is a 3-step filtration such thatFil0(N) := 0,gr1(N) = Fil1(N) = Ker(εN), gr2(N) = Ker(A1 ⊗A0 N → A3 ⊗A0 N) and gr3(N) =A2 ⊗A0 N . By an easy devissage, we reduce to verify thatalExtiA0

(P, grj(N)) = 0 for everyi > 0 andj = 1, 2, 3. However,gr2(N) is anA1-module andgr3(N) is anA2-module, so therequired vanishing follows forj = 2, 3. Moreover, applying− ⊗A0 N to (3.3.11), we derive ashort exact sequence :

0→ TorA01 (N,A2)→

TorA01 (N,A3)

TorA01 (N,A1)

→ gr1(N)→ 0.(3.3.16)

Here again, the leftmost term of (3.3.16) is anA2-module, and the middle term is anA1-module,so the same devissage yields the sought vanishing also forj = 1.

Corollary 3.3.17. In the situation of(3.3.8), denote byAi,fl (resp.Eti, resp.w.Eti) the cate-gory of flat (resp.etale, resp. weaklyetale)Ai-algebras. The functorπ induces equivalences

A0,fl∼→ A1,fl×A3,fl

A2,fl Et0∼→ Et1×Et3

Et2 w.Et0∼→ w.Et1×w.Et3

w.Et2.


3.3.3.3.18. Next we want to reinterpret the equivalences ofproposition 3.3.15 in terms ofdescent data. IfF : C → V a-Algo is a fibred category over the opposite of the category ofalmost algebras, and ifX → Y is a given morphism of almost algebras, we shall denote byDesc(C , Y/X) the category of objects of the fibre categoryFY , endowed with a descent datumrelative to the morphismX → Y (cp. [28, Ch.II§1]). In the arguments hereafter, we considermorphisms of almost algebras and modules, and one has to reverse the direction of the arrowsto pass to morphisms in the relevant fibred category. Denote by pi : Y → Y ⊗X Y (i = 1, 2),resp.pij : Y ⊗X Y → Y ⊗X Y ⊗X Y (1 ≤ i < j ≤ 3) the usual morphisms.

3.3.3.3.19. As an example,Desc(V a-Alg.Modo, Y/X) consists of the pairs(M,β) whereM is aY -module andβ is aY ⊗X Y -linear isomorphismβ : p∗2(M)

∼→ p∗1(M) such that

p∗12(β) p∗23(β) = p∗13(β).(3.3.20)

3.3.3.3.21. Let nowI ⊂ X be an ideal, and setX := X/I, Y := Y/I · Y . For anyF : C →V a-Algo as above, one has an essentially commutative diagram:

Desc(C , Y/X) //

Desc(C , Y /X)

FY // FY .

This induces a functor :

Desc(C , Y/X)→ Desc(C , Y /X)×FYFY .(3.3.22)

Lemma 3.3.23.With the above notation, suppose moreover that the natural morphismI →I · Y is an isomorphism. Then the functor(3.3.22)is an equivalence wheneverC is one of thefibred categoriesV a-Alg.Modo, V a-Alg.Morpho, Et

o, w.Et

o.

Proof. For anyn > 0, denote byY ⊗n (resp.Y⊗n

) then-fold tensor product ofY (resp.Y ) withitself overX (resp.X), and byρn : Y ⊗n → Y

⊗nthe natural morphism. First of all we claim

that, for everyn > 0, the natural diagram of almost algebras

Y ⊗nρn //

µn

Y⊗n

µn

Y

ρ1 // Y

(3.3.24)

is cartesian (whereµn andµn aren-fold multiplication morphisms). For this, we need to verifythat, for everyn > 0, the induced morphismKer(ρn) → Ker(ρ1) (defined by multiplicationof the first two factors) is an isomorphism. It then suffices tocheck that the natural morphismKer(ρn) → Ker(ρn−1) is an isomorphism for alln > 1. Indeed, consider the commutativediagram

I ⊗X Y ⊗n−1p // I · Y ⊗n−1

i //

ψ

Y ⊗n−1

φ⊗1Y⊗n−1

1Y ⊗n−1

((QQQQQQQQQQQQQ

I ⊗X Y ⊗n−1 p′// Ker(ρn)

i′// Y ⊗n

µY/X⊗1Y ⊗n−2

// Y ⊗n−1.

FromI ·Y = φ(Y ), it follows thatp′ is an epimorphism. Hence alsoψ is an epimorphism. Sincei is a monomorphism, it follows thatψ is also a monomorphism, henceψ is an isomorphismand the claim follows easily.

We consider first the caseC := V a-Alg.Modo; we see that (3.3.24) is a diagram of the kindconsidered in (3.3.8), hence, for everyn > 0, we have the associated functorπn : Y ⊗n-Mod→


Y⊗n

-Mod ×Y -Mod Y -Mod and also its right adjointTn. Denote bypi : Y → Y⊗2

(i =

1, 2) the usual morphisms, and similarly definepij : Y⊗2→ Y

⊗3. Suppose there is given

a descent datum(M,β) for M , relative toX → Y . The cocycle condition (3.3.20) implieseasily thatµ∗2(β) is the identity onµ∗2(p

∗i (M)) = M . It follows that the pair(β, 1M) defines

an isomorphismπ2(p∗1M)∼→ π2(p

∗2M) in the categoryY

⊗2-Mod ×Y -Mod Y -Mod. Hence

T2(β, 1M) : T2 π2(p∗1M) → T2 π2(p∗2M) is an isomorphism. However, we remark thateither morphismpi yields a section forµ2, hence we are in the situation contemplated in lemma3.3.12, and we derive an isomorphismβ : p∗2(M)

∼→ p∗1(M). We claim that(M,β) is an

object ofDesc(C , Y/X), i.e. that β verifies the cocycle condition (3.3.20). Indeed, we cancompute:π3(p∗ijβ) = (ρ∗3(p

∗ijβ), µ

∗3(p∗ijβ)) and by construction we haveρ∗3(p

∗ijβ) = p∗ij(β) and

µ∗3(p∗ijβ) = µ∗2(β) = 1M . Therefore, the cocycle identity forβ implies the equalityπ3(p∗12(β))

π3(p∗23(β)) = π3(p

∗13(β)). If we now apply the functorT3 to this equality, and then invoke again

lemma 3.3.12, the required cocycle identity forβ will ensue. Clearlyβ is the only descent datumonM lifting β. This proves that (3.3.22) is essentially surjective. The same sort of argumentalso shows that the functor (3.3.22) induces bijections on morphisms, so the lemma follows inthis case. Next, the caseC := V a-Alg.Morpho can be deduced formally from the previouscase, by applying repeatedly natural isomorphisms of the kindp∗i (M ⊗Y N) ≃ p∗i (M)⊗Y⊗XY

p∗i (N) (i = 1, 2). Finally, the “etaleness” of an object ofDesc(V a-Alg.Morpho, Y/X) canbe checked on its projection ontoY -Algo, hence also the casesC := w.Et

oandC := Et

o

follow directly.

3.3.3.3.25. Now, letB := A1 × A2; to an object(M,β) in Desc(V a-Alg.Modo, B/A) weassign an object(M1,M2, ξ) of M1 ×M3 M2, as follows. SetMi := Ai ⊗B M (i = 1, 2) andAij := Ai ⊗A0 Aj . We can writeB ⊗A0 B =

∏2i,j=1Aij andβ gives rise to theAij-linear

isomorphismsβij : Aij ⊗B⊗A0B p∗2(M)

∼→ Aij ⊗B⊗A0

B p∗1(M). In other words, we obtainisomorphismsβij : Ai ⊗A0 Mj → Mi ⊗A0 Aj. However, we have a natural isomorphismA12 ≃ A3 (indeed, suppose thatA1 → A3 is an epimorphism with kernelI; thenI is also anideal ofA0 andA0/I ≃ A2; now the claim follows by remarking thatI · A1 = I). Hence wecan chooseξ = β12. In this way we obtain a functor :

Desc(V a-Alg.Modo, B/A0)→ (M1 ×M3 M2)o.(3.3.26)

Proposition 3.3.27.The functor(3.3.26)is an equivalence of categories.

Proof. Let us say thatA1 → A3 is an epimorphism with kernelI. ThenI is also an idealof B and we haveB/I ≃ A3 × A2 andA0/I ≃ A2. We intend to apply lemma 3.3.23 tothe morphismA0 → B. However, the induced morphismB := B/I → A0 := A0/I inV a-Algo has a section, and hence it is of universal effective descentfor every fibred category.Thus, we can replace in (3.3.22) the categoryDesc(V a-Alg.Modo, B/A0) byA0-Modo, andthereby, identify (up to equivalence) the target of (3.3.22) with the 2-fibred product(M1 ×M2)

o ×(M3×M2)o M o2 . The latter is equivalent to the categoryM o

1 ×M o3

M o2 and the resulting

functorDesc(V a-Alg.Modo, B/A0) →M o1 ×M o

3M o

2 is canonically isomorphic to (3.3.26),which gives the claim.

Putting together propositions 3.3.15 and 3.3.27 we obtain the following :

Corollary 3.3.28. In the situation of(3.3.8), the morphismA0 → A1×A2 is of effective descentfor the fibred categories of flat modules and of almost projective modules.

3.3.3.3.29. Next we would like to give sufficient conditionsto ensure that a morphism ofalmost algebras is of effective descent for the fibred category w.Et

o→ V a-Algo of weakly

etale algebras (resp. for etale algebras). To this aim we are led to the following :


Definition 3.3.30. A morphismφ : A → B of almost algebras is said to bestrictly finite ifKer(φ) is nilpotent andB ≃ Ra, whereR is a finiteA∗-algebra.

Theorem 3.3.31.Letφ : A→ B be a strictly finite morphism of almost algebras. Then :

(i) For everyA-algebraC, the induced morphismC → C ⊗A B is again strictly finite.(ii) If M is a flatA-module andB ⊗A M is almost projective overB, thenM is almost

projective overA.(iii) A → B is of universal effective descent for the fibred categories of weaklyetale (resp.

etale) almost algebras.

Proof. (i): suppose thatB = Ra for a finiteA∗-algebraR; thenS := C∗ ⊗A∗ R is a finiteC∗-algebra andSa ≃ C. It remains to show thatKer(C → C ⊗A B) is nilpotent. Suppose thatRis generated byn elements as anA∗-module and letFA∗(R) (resp.FC∗(S)) be the Fitting idealof R (resp.ofS); we haveAnnC∗(S)

n ⊂ FC∗(S) ⊂ AnnC∗(S) (see [37, Ch.XIX Prop.2.5]); onthe other handFC∗(S) = FA∗(R) · C∗, so the claim is clear.

(iii): we shall consider the fibred categoryF : w.Eto→ V a-Algo; the same argument

applies also to etale almost algebras. We begin by establishing a very special case :

Claim3.3.32. Assertion (iii) holds whenB = (A/I1) × (A/I2), whereI1 andI2 are ideals inA such thatI1 ∩ I2 is nilpotent.

Proof of the claim:First of all we remark that the situation considered in the claim is stableunder arbitrary base change, therefore it suffices to show thatφ is of F -2-descent in this case.Then we factorφ as a compositionA → A/Ker(φ) → B and we remark thatA → A/Ker(φ)is ofF -2-descent by theorem 3.2.17; since a composition of morphisms ofF -2-descent is againof F -2-descent, we are reduced to show thatA/Ker(φ) → B is of F -2-descent,i.e. we canassume thatKer(φ) ≃ 0. However, in this case the claim follows easily from corollary 3.3.28.

Claim3.3.33. More generally, assertion (iii) holds whenB =∏n

i=1A/Ii, whereI1, ..., In areideals ofA, such that

⋂ni=1 Ii is nilpotent.

Proof of the claim:We prove this by induction onn, the casen = 2 being covered by claim3.3.32. Therefore, suppose thatn > 2, and setB′ := A/(

⋂n−1i=1 Ij). By induction, the morphism

B′ →∏n−1

i=1 A/Ii is of universalF -2-descent. However, according to [28, Ch.II Prop.1.1.3],the sieves of universalF -2-descent form a topology onV a-Algo; for this topology,A,B is acovering family ofA×B and(A→ B′× (A/In))

o is a covering morphism, henceB′, A/Inis a covering family ofA, and then, by composition of covering families,

∏n−1i=1 A/Ii, A/In

is a covering family ofA, which is equivalent to the claim.Now, let A → B be a general strictly finite morphism, so thatB = Ra for some fi-

nite A∗-algebraR. Pick generatorsf1, ..., fm of theA∗-moduleR, and monic polynomialsp1(X), ..., pm(X) such thatpi(fi) = 0 for i = 1, ..., m.

Claim3.3.34. There exists a finite and faithfully flat extensionC of A∗ such that the images inC[X ] of p1(X),...,pm(X) split as products of monic linear factors.

Proof of the claim:This extensionC can be obtained as follows. It suffices to find, for eachi = 1, ..., m, an extensionCi that splitspi(X), because thenC := C1 ⊗A∗ ...⊗A∗ Cm will splitthem all, so we can assume thatm = 1 andp1(X) = p(X); moreover, by induction on thedegree ofp(X), it suffices to find an extensionC ′ such thatp(X) factors inC ′[X ] as a productof the formp(X) = (X − α) · q(X), whereq(X) is a monic polynomial of degreedeg(p)− 1.Clearly we can takeC ′ := A∗[T ]/(p(T )).

Given aC as in claim 3.3.34, we remark that the morphismA → Ca is of universalF -2-descent. Considering again the topology of universalF -2-descent, it follows thatA → B


is of universalF -2-descent if and only if the same holds for the induced morphism Ca →Ca ⊗A B. Therefore, in proving assertion (iii) we can replaceφ by 1C ⊗A φ and assumefrom start that the polynomialspi(X) factor inA∗[X ] as product of linear factors. Now, letdi := deg(pi) andpi(X) :=

∏dij (X−αij) (for i = 1, ..., m). We get a surjective homomorphism

of A∗-algebrasD := A∗[X1, ..., Xm]/(p1(X1), ..., pm(Xm)) → R by the ruleXi 7→ fi (i =1, ..., m). Moreover, any sequenceα := (α1,j1, α2,j2, ..., αm,jm) yields a homomorphismψα :D → A∗, determined by the assignmentXi 7→ αi,ji. A simple combinatorial argument showsthat

∏αKer(ψα) = 0, whereα runs over all the sequences as above. Hence the product map∏

α ψα : D →∏

αA∗ has nilpotent kernel. We notice that theA∗-algebra(∏

αA∗) ⊗D R is aquotient of

∏αA∗, hence it can be written as a product of rings of the formA∗/Iα, for various

idealsIα. By (i), the kernel of the induced homomorphismR →∏

αA∗/Iα is nilpotent, hencethe same holds for the kernel of the compositionA→

∏αA/I

aα, which is therefore of the kind

considered in claim 3.3.33. HenceA →∏

αA/Iaα is of universalF -2-descent. Since such

morphisms form a topology, it follows that alsoA → B is of universalF -2-descent, whichconcludes the proof of (iii).

Finally, letM be as in (ii) and pick againC as in the proof of claim 3.3.34. By remark3.2.23(iv),M is almost projective overA if and only ifCa ⊗AM is almost projective overCa;hence we can replaceφ by 1Ca ⊗A φ, and by arguing as in the proof of (iii), we can assumefrom start thatB =

∏nj=1(A/Ij) for idealsIj ⊂ A, j = 1, ..., n such thatI :=

⋂nj=1 Ij is

nilpotent. By an easy induction, we can furthermore reduce to the casen = 2. We factorφ asA → A/I → B; by proposition 3.3.15 it follows that(A/I) ⊗A M is almost projective overA/I, and then lemma 3.2.22(i) says thatM itself is almost projective.

Remark 3.3.35. It is natural to ask whether theorem 3.3.31 holds if we replace everywhere“strictly finite” by “finite with nilpotent kernel” (or even by “almost finite with nilpotent ker-nel”). We do not know the answer to this question.

3.3.3.3.36. On the categoryV a-Alg (taken in some universe) consider the topologiesτe (resp.τw) of universal effective descent for the fibred categoryEt

o(resp.w.Et

o). For a ringR denote

by Idemp(R) the set of idempotents ofR.

Proposition 3.3.37.With the notation of(3.3.3.3.36)we have:

(i) The presheafA 7→ Idemp(A∗) is a sheaf for bothτe andτw.(ii) If f : A→ B is anetale (resp. weaklyetale) morphism of almostV -algebras and there is

a covering family(A → Aα)o for τe (resp.τw) such thatAα → Aα ⊗A B is an almost

projective epimorphism for allα, thenf is an almost projective epimorphism.(iii) τe is finer thanτw.

Proof. (i): use descent of morphisms and the bijection

HomA-Alg(A×A,A)∼→ Idemp(A∗) φ 7→ φ∗(1, 0).

(ii): by remark 3.1.8,Ker(Aα → Aα⊗AB) is generated byeα ∈ Idemp(Aα∗). eα andeβ agreein (Aα ⊗A Aβ)∗, so by (i) there is an idempotente ∈ A∗ that restricts toeα in Idemp(Aα∗),for eachα. TheA-algebrasB andA/e · A become isomorphic after applying− ⊗A Aα; theseisomorphisms are unique and are compatible onAα⊗AAβ, hence they patch to an isomorphismB ≃ A/e ·A.

(iii): we have to show that ifA is an almostV -algebra,R a sieve of(V a-Alg)o/A andRis of universalw.Et

o-2-descent, thenR is of universalEt

o-2-descent. Since the assumption is

stable under base change, it suffices to show thatR is of Eto-2-descent. Descent of morphisms

is clear. LetR be the sieve generated by a family of morphisms(A → Aα)o. Any descent

datum consisting of etaleAα-algebrasBα and isomorphismsAα⊗ABβ ≃ Bα⊗AAβ satisfying


the cocycle condition, becomes effective when we pass tow.Eto. So one has to verify that ifB

is a weakly etaleA-algebra such thatB ⊗A Aα is etale overAα for all α, thenB is etale overA. Indeed, an application of (ii) gives thatB ⊗A B → B is almost projective.

3.3.3.3.38. We conclude with a digression to explain the relationship between our results andrelated facts that can be extracted from the literature. So,we now place ourselves in the “clas-sical limit” m := V (cp. example 2.1.2(ii)). In this case, weakly etale morphisms had alreadybeen considered in some earlier work, and they were called “absolutely flat” morphisms. A ringhomomorphismA→ B is etale in the usual sense of [29] if and only if it is absolutely flat andof finite presentation. Let us denote byu.Et

othe fibred category overV -Algo, whose fibre

over aV -algebraA is the opposite of the category of etaleA-algebras in the usual sense. Weclaim that, if a morphismA → B of V -algebras is of universal effective descent for the fibredcategoryw.Et

o(resp. Et

o), then it is a morphism of universal effective descent foru.Et

o.

Indeed, letC be an etaleA-algebra (in the sense of definition 3.1.1) and such thatC ⊗A B isetale overB in the usual sense. We have to show thatC is etale in the usual sense,i.e. that it isof finite presentation overA. This amounts to showing that, for every filtered inductive system(Aλ)λ∈Λ of A-algebras, we havecolim

λ∈ΛHomA-Alg(C,Aλ) ≃ HomA-Alg(C, colim

λ∈ΛAλ). Since, by

assumption, this is known after extending scalars toB and toB ⊗A B, it suffices to show that,for anyA-algebraD, the natural sequence

HomA-Alg(C,D) // HomB-Alg(CB, DB)//// HomB⊗AB-Alg(CB⊗AB, DB⊗AB)

is exact. For this, note thatHomA-Alg(C,D) = HomD-Alg(CD, D) (and similarly for the otherterms) and by hypothesis(D → D ⊗A B)o is a morphism of 1-descent for the fibred categoryw.Et

o(resp.Et

o).

As a consequence of these observations and of theorem 3.3.31, we see that any finite ringhomomorphismφ : A→ B with nilpotent kernel is of universal effective descent forthe fibredcategory of etale algebras. This fact was known as follows.By [29, Exp.IX, 4.7],Spec(φ) is ofuniversal effective descent for the fibred category of separated etale morphisms of finite type.One has to show that ifX is such a scheme overA, such thatX⊗AB is affine, thenX is affine.This follows by reduction to the noetherian case and [19, Ch.II, 6.7.1].

3.4. Behaviour of etale morphisms under Frobenius.We consider the following categoryBof base rings. The objects ofB are the pairs(V,m), whereV is a ring andm is an ideal ofVwith m = m2 andm is flat. The morphisms(V,mV )→ (W,mW ) between two objects ofB arethe ring homomorphismsf : V →W such thatmW = f(mV ) ·W .

3.4.3.4.1. We have a fibred and cofibred categoryB-Mod → B (see [29, Exp.VI§5,6,10]for generalities on fibred categories). An object ofB-Mod (which we may call a “B-module”)consists of a pair((V,m),M), where(V,m) is an object ofB andM is aV -module. Giventwo objectsX := ((V,mV ),M) andY := ((W,mW ), N), the morphismsX → Y are the pairs(f, g), wheref : (V,mV )→ (W,mW ) is a morphism inB andg :M → N is anf -linear map.

3.4.3.4.2. Similarly one has a fibred and cofibred categoryB-Alg → B of B-algebras.We will also need to consider the fibred and cofibred categoryB-Mon → B of non-unitarycommutativeB-monoids: an object ofB-Mon is a pair((V,m), A) whereA is aV -moduleendowed with a morphismA⊗V A→ A subject to associativity and commutativity conditions,as discussed in section 2.2. The fibre over an object(V,m) of B, is the category ofV -monoidsdenoted(V,m)-Mon or simplyV -Mon.


3.4.3.4.3. The almost isomorphisms in the fibres ofB-Mod → B give a multiplicativesystemΣ in B-Mod, admitting a calculus of both left and right fractions. The “locallysmall” conditions are satisfied (see [50, p.381]), so that one can form the localised categoryBa-Mod := Σ−1(B-Mod). The fibres of the localised category over the objects ofB are thepreviously considered categories of almost modules. Similar considerations hold forB-Alg

andB-Mon, and we get the fibred and cofibred categoriesBa-Mod → B, Ba-Alg → B

andBa-Mon→ B. In particular, for every object(V,m) of B, we have an obvious notion ofalmostV -monoid and the category consisting of these is denotedV a-Mon.

3.4.3.4.4. The localisation functors

B-Mod→ Ba-Mod : M 7→Ma

B-Alg→ Ba-Alg : B 7→ Ba

have left and right adjoints. These adjoints can be chosen asfunctors of categories overB suchthat the adjunction units and counits are morphisms over identity arrows inB. On the fibresthese induce the previously considered left and right adjointsM 7→ M!, M 7→ M∗, B 7→ B!!,B 7→ B∗. We will use the same notation for the corresponding functors on the larger categories.Then it is easy to check that the functorM 7→ M! is cartesian and cocartesian (i.e. it sendscartesian arrows to cartesian arrows and cocartesian arrows to cocartesian arrows),M 7→ M∗andB 7→ B∗ are cartesian, andB 7→ B!! is cocartesian.

3.4.3.4.5. LetB/Fp be the full subcategory ofB consisting of all objects(V,m) whereV is anFp-algebra. Define similarlyB-Alg/Fp, B-Mon/Fp andBa-Alg/Fp, Ba-Mon/Fp, so thatwe have again fibred and cofibred categoriesBa-Alg/Fp → B/Fp andBa-Alg/Fp → B/Fp(resp. the same for non-unitary monoids). We remark that thecategoriesBa-Alg/Fp andBa-Mon/Fp have small limits and colimits, and these are preserved by the projection toB/Fp.Especially, ifA→ B andA→ C are two morphisms inBa-Alg/Fp or Ba-Mon/Fp, we candefineB ⊗A C as such a colimit.

3.4.3.4.6. IfA is a (unitary or non-unitary)B-monoid overFp, we denote byΦA : A→ A theFrobenius endomorphism: x 7→ xp. If (V,m) is an object ofB/Fp, it follows from proposition2.1.7(ii) thatΦV : (V,m)→ (V,m) is a morphism inB. If B is an object ofB-Alg/Fp (resp.B-Mon/Fp) overV , then the Frobenius map induces a morphismΦB : B → B in B-Alg/Fp(resp. B-Mon/Fp) overΦV . In this way we get a natural transformation from the identityfunctor ofB-Alg/Fp (resp.B-Mon/Fp) to itself that induces a natural transformation on theidentity functor ofBa-Alg/Fp (resp.Ba-Mon/Fp).

3.4.3.4.7. Using the pull-back functors, any objectB of B-Alg overV defines new objectsB(m) of B-Alg (m ∈ N) overV , whereB(m) := (ΦmV )

∗(B), which is justB considered as

a V -algebra via the homomorphismVΦm

−→ V → B. These operations also induce functorsB 7→ B(m) on almostB-algebras.

Definition 3.4.8. Let (V,m) be an object ofB/Fp.

(i) We say that a morphismf : A → B of almostV -algebras (resp. almostV -monoids) isinvertible up toΦm if there exists a morphismf ′ : B → A in Ba-Alg (resp.Ba-Mon)overΦmV , such thatf ′ f = ΦmA andf f ′ = ΦmB .

(ii) We say that an almostV -monoidI (e.g.an ideal in aV a-algebra) isFrobenius nilpotentifΦI is nilpotent.

3.4.3.4.9. Notice that a morphismf of V a-Alg (or V a-Mon) is invertible up toΦm if andonly if f∗ : A∗ → B∗ is so as a morphism ofFp-algebras.

Lemma 3.4.10.Let(V,m) be an object ofB/Fp and letf : A→ B, g : B → C be morphismsof almostV -algebras or almostV -monoids.


(i) If f (resp.g) is invertible up toΦn (resp.Φm), theng f is invertible up toΦm+n.(ii) If f (resp.g f ) is invertible up toΦn (resp.Φm), theng is invertible up toΦm+n.(iii) If g (resp.g f ) is invertible up toΦn (resp.Φm), thenf is invertible up toΦm+n.(iv) The Frobenius morphisms induceΦV -linear morphisms (i.e. morphisms inBa-Mod over

ΦV ) Φ′ : Ker(f) → Ker(f) andΦ′′ : Coker(f) → Coker(f), andf is invertible up tosome power ofΦ if and only if bothΦ′ andΦ′′ are nilpotent.

(v) Consider a map of short exact sequences of almostV -monoids :

0 // A′ //

f ′

A //

f

A′′ //

f ′′

0

0 // B′ // B // B′′ // 0

and suppose that two of the morphismsf ′, f, f ′′ are invertible up to a power ofΦ. Thenalso the third morphism has this property.

Proof. (i): if f ′ is an inverse off up toΦn andg′ is an inverse ofg up toΦm, thenf ′ g′ isan inverse ofg f up toΦm+n. (ii): given an inversef ′ of f up toΦn and an inverseh′ ofh := g f up toΦm, let g′ := ΦnB f h

′. We compute :

g g′ = g ΦnB f h′ = ΦnC g f h = ΦnC Φ

mC

g′ g = ΦnB f h′ g = f h′ g ΦnB = f h′ g f f ′

= f ΦmA f′ = ΦmB f f

′ = ΦmB ΦnB.

(iii) is similar and (iv) is an easy diagram chasing left to the reader. (v) follows from (iv) andthe snake lemma.

Lemma 3.4.11.Let (V,m) be an object ofB/Fp.

(i) If f : A → B is a morphism of almostV -algebras, invertible up toΦn, then so isA′ →A′ ⊗A B for every morphismA→ A′ of almost algebras.

(ii) If f : (V,mV )→ (W,mW ) is a morphism inB/Fp, the functors

f∗ : (V,mV )a-Alg→ (W,mW )a-Alg and f ∗ : (W,mW )a-Alg→ (V,mV )

a-Alg

preserve the class of morphisms invertible up toΦn.

Proof. (i): givenf ′ : B → A(m), construct a morphismA′ ⊗A B → A′(m) using the morphismA′ → A′(m) coming fromΦmA′ andf ′. (ii): the assertion forf ∗ is clear, and the assertion forf∗follows from (i).

Remark 3.4.12. Statements like those of lemma 3.4.11 hold for the classes offlat, (weakly)unramified, (weakly) etale morphisms.

Theorem 3.4.13.Let (V,m) be an object ofB/Fp andf : A→ B a weaklyetale morphism ofalmostV -algebras.

(i) If f is invertible up toΦn (n ≥ 0), then it is an isomorphism.(ii) For every integerm ≥ 0 the natural square diagram

Af //

ΦmA

B

ΦmB

A(m)

f(m) // B(m)

(3.4.14)

is cocartesian.


Proof. (i): we first show thatf is faithfully flat. Sincef is flat, it remains to show that ifMis anA-module such thatM ⊗A B = 0, thenM = 0. It suffice to do this forM := A/I,for an arbitrary idealI of A. After base change byA → A/I, we reduce to show thatB = 0impliesA = 0. However,A∗ → B∗ is invertible up toΦn, soΦnA∗

= 0 which meansA∗ = 0. Inparticular,f is a monomorphism, hence the proof is complete in case thatf is an epimorphism.

In general, consider the compositionB1B⊗f−→ B ⊗A B

µB/A−→ B. From lemma 3.4.11(i) it follows

that1B⊗f is invertible up toΦn; then lemma 3.4.10(ii) says thatµB/A is invertible up toΦn. Thelatter is also weakly etale; by the foregoing we derive thatit is an isomorphism. Consequently1B ⊗ f is an isomorphism, and finally, by faithful flatness,f itself is an isomorphism.

(ii): the morphismsΦmA andΦmB are invertible up toΦm. By lemma 3.4.11(i) it follows that1B ⊗ ΦmA : B → B ⊗A A(m) is invertible up toΦm; hence, by lemma 3.4.10(ii), the morphismh : B⊗AA(m) → B(m) induced byΦmB andf(m) is invertible up toΦ2m (in fact one verifies thatit is invertible up toΦm). But h is a morphism of weakly etaleA(m)-algebras, so it is weaklyetale, so it is an isomorphism by (i).

Remark 3.4.15. Theorem 3.4.13(ii) extends a statement of Faltings ([24, p.10]) for his notionof almost etale extensions.

3.4.3.4.16. We recall (cp. [28, Ch.0, 3.5]) that a morphismf : X → Y of objects in a siteis calledbicoveringif the induced map of associated sheaves of sets is an isomorphism; if f issquarable (“quarrable” in French), this is equivalent to the condition that bothf and the diagonalmorphismX → X ×Y X are covering morphisms.

3.4.3.4.17. LetF → E be a fibered category andf : P → Q a squarable morphism ofE.Consider the following condition:

for every base changeP×QQ′ → Q′ of f , the inverse image functorFQ′ → FP×QQ′

is an equivalence of categories.(3.4.18)

Inspecting the arguments in [28, Ch.II,§1.1] one can show:

Lemma 3.4.19.With the above notation, letτ be the topology of universal effective descentrelative toF → E. Then we have :

(i) if (3.4.18)holds, thenf is a covering morphism for the topologyτ .(ii) f is bicovering forτ if and only if (3.4.18)holds both forf and for the diagonal morphism

P → P ×Q P .

Remark 3.4.20. In [28, Ch.II, 1.1.3(iv)] it is stated that “la reciproque est vraie sii = 2”,meaning that (3.4.18) is equivalent to the condition thatf is bicovering forτ . (Actually thecited statement is given in terms of presheaves, but one can show that (3.4.18) is equivalent tothe corresponding condition for the fibered categoryF+ → EU considered inloc.cit.) However,this fails in general : as a counterexample we can give the following. LetE be the category ofschemes of finite type over a fieldk; setP = A1

k, Q = Spec(k). Finally letF → E be thediscretely fibered category defined by the presheafX 7→ H0(X,Z). Then it is easy to showthatf satisfies (3.4.18) but the diagonal map does not, sof is not bicovering. The mistake inthe proof is in [28, Ch.II, 1.1.3.5], where one knows thatF+(d) is an equivalence of categories(notation ofloc.cit.) but one needs it also after base changes ofd.

Lemma 3.4.21. (i) Let f : A→ B be a morphism ofV a-algebras.

(i) If f is invertible up toΦm, then the induced functorsEt(A) → Et(B) andw.Et(A) →w.Et(B) are equivalences of categories.

(ii) If f is weaklyetale andC → D is a morphism ofA-algebras invertible up toΦm, then theinduced map:HomA-Alg(B,C)→ HomA-Alg(B,D) is bijective.


Proof. We first consider (i) for the special case wheref := ΦmA : A → A(m). The functor(ΦmV )

∗ : V a-Alg→ V a-Alg induces a functor(−)(m) : A-Alg→ A(m)-Alg, and by restriction(see remark 3.4.11) we obtain a functor(−)(m) : Et(A) → Et(A(m)); by theorem 3.4.13(ii),the latter is isomorphic to the functor(Φm)∗ : Et(A)→ Et(A(m)) of the lemma. Furthermore,from remark 2.1.4(ii) and (2.2.4) we derive a natural ring isomorphismω : A(m)∗ ≃ A∗, hencean essentially commutative diagram

Et(A) //

(Φm)∗

A-Algα //

(−)(m)

(A∗,m · A∗)a-Alg

ω∗

Et(A(m)) // A(m)-Algβ // (A(m)∗,m · A(m)∗)

a-Alg

whereα andβ are the equivalences of remark 2.2.12. Clearlyα andβ restrict to equivalenceson the corresponding categories of etale algebras, hence the lemma follows in this case.

For the general case of (i), letf ′ : B → A(m) be a morphism as in definition 3.4.8. Dia-gram (3.4.14) induces an essentially commutative diagram of the corresponding categories ofalgebras, so by the previous case, the functor(f ′)∗ : Et(B) → Et(A(m)) has both a left es-sential inverse and a right essential inverse; these essential inverses must be isomorphic, sof∗has an essential inverse as desired. Finally, we remark thatthe map in (ii) is the same as themapHomC-Alg(B ⊗A C,C)→ HomD-Alg(B ⊗A D,D), and the latter is a bijection in view of(i).

Remark 3.4.22. Notice that lemma 3.4.21(ii) generalises the lifting theorem 3.2.17(i) (in caseV is anFp-algebra). Similarly, it follows from lemmata 3.4.21(i) and 3.4.10(iv) that, in caseVis anFp-algebra, one can replace “nilpotent” in theorem 3.2.17 parts (ii) and (iii) by “Frobeniusnilpotent”.

3.4.3.4.23. In the following,τ will denote indifferently the topology of universal effectivedescent defined by either of the fibered categoriesw.Et

o→ V a-Algo or Et

o→ V a-Algo.

Proposition 3.4.24. If f : A → B is a morphism of almostV -algebras which is invertible uptoΦm, thenf o is bicovering for the topologyτ .

Proof. In light of lemmata 3.4.19(ii) and 3.4.21(i), it suffices to show thatµB/A is invertible up

to a power ofΦ. For this, factor the identity morphism ofB asB1B⊗f−→ B ⊗A B

µB/A−→ B and

argue as in the proof of theorem 3.4.13.

Proposition 3.4.25.LetA→ B be a morphism of almostV -algebras andI ⊂ A an ideal. SetA := A/I andB := B/I · B. Suppose that either

(a) I → I · B is an epimorphism with nilpotent kernel, or(b) V is anFp-algebra andI → I · B is invertible up to a power ofΦ.

Then we have :

(i) conditions(a) and(b) are stable under any base changeA→ C.(ii) (A→ B)o is covering (resp. bicovering) forτ if and only if(A→ B)o is.

Proof. Suppose first thatI → I · B is an isomorphism; in this case we claim thatI · C →I · (C ⊗A B) is an epimorphism andKer(I · C → I · (C ⊗A B))2 = 0 for anyA-algebraC.Indeed, since by assumptionI ≃ I · B, C ⊗A B acts onC ⊗A I, henceKer(C → C ⊗A B)annihilatesC ⊗A I, hence annihilates its imageI · C, whence the claim. If, moreover,V is anFp-algebra, lemma 3.4.10(iv) implies thatI · C → I · (C ⊗A B) is invertible up to a power ofΦ.


In the general case, consider the intermediate almostV -algebraA1 := A ×B B equippedwith the idealI1 := 0×B (I ·B). In case (a),I1 = I · A1 andA→ A1 is an epimorphism withnilpotent kernel, hence it remains such after any base changeA→ C. To prove (i) in case (a), itsuffices then to consider the morphismA1 → B, hence we can assume from start thatI → I ·Bis an isomorphism, which is the case already dealt with. To prove (i) in case (b), it suffices toconsider the cases of(A, I) → (A1, I1) and(A1, I1) → (B, I · B). The second case is treatedabove. In the first case, we do not necessarily haveI1 = I · A1 and the assertion to be checkedis that, for everyA-algebraC, the morphismI ·C → I1 · (A1⊗A C) is invertible up to a powerof Φ. We apply lemma 3.4.10(v) to the commutative diagram with exact rows:

0 // I //

A //

A/I // 0

0 // I · B // A1// A/I // 0

to deduce thatA→ A1 is invertible up to some power ofΦ, hence so isC → A1 ⊗A C, whichimplies the assertion.

As for (ii), we remark that the “only if” part is trivial; and we assume therefore that(A→ B)o

is τ -covering (resp.τ -bicovering). Consider first the assertion for “covering”.We need to showthat(A→ B)o is of universal effective descent forF , whereF is either one of our two fiberedcategories. In light of (i), this is reduced to the assertionthat(A → B)o is of effective descentfor F . We notice that(A → A1)

o is bicovering forτ (in case (a) by theorem 3.2.17 andlemma 3.4.19(ii), in case (b) by proposition 3.4.24). As(A → A1/I1)

o is an isomorphism, theassertion is reduced to the case whereI → I · B is an isomorphism. In this case, by lemma3.3.23, there is a natural equivalence:Desc(F,B/A)

∼→ Desc(F,B/A) ×FB

FB. Then theassertion follows easily from corollary 3.3.17. Finally suppose that(A → B)o is bicovering.The foregoing already says that(A→ B)o is covering, so it remains to show that(B ⊗A B →B)o is also covering. The above argument again reduces to the case whereI → I · B is anisomorphism. Then, as in the proof of lemma 3.3.23, the induced morphismI ·(B⊗AB)→ I ·Bis an isomorphism as well. Thus the assertion for “bicovering” is reduced to the assertion for“covering”.

We conclude this section with a result of a more special nature, which can be interpreted asan easy case of almost purity in positive characteristic.

3.4.3.4.26. We suppose now that the basic setup(V,m) consists of aperfectFp-algebraV ,i.e. such that the Frobenius endomorphismΦV : V → V is bijective; moreover we assumethat there exists a non-zero-divisorε ∈ m such thatm =

⋃n>0 ε

1/pm · V . Let us denote byEt(V a)uafp (resp.u.Et(V [ε−1])fp) the category of uniformly almost finite projective etaleV a-algebras (resp. of finite etaleV [ε−1]-algebra in the usual sense of [29]). We will be concernedwith the natural functor:

Et(V a)uafp → u.Et(V [ε−1])fp : A 7→ A∗[ε−1].(3.4.27)

Theorem 3.4.28.Under the assumptions of(3.4.3.4.26), the functor(3.4.27)is an equivalenceof categories.

Proof. Let R be a finite etaleV [ε−1]-algebra. SinceV [ε−1] is perfect, the same holds forR,in view of theorem 3.4.13(ii) (applied in the classical limit case of example 2.1.2(ii)). Let uschoose a finiteV -algebraR0 ⊂ R such thatR0[ε

−1] = R and defineR1 :=⋃n∈N Φ

−nR (R0).

Claim 3.4.29. TheV a-algebraRa1 does not depend on the choice ofR0.


Proof of the claim:LetR′0 ⊂ R be another finiteV -algebra such thatR′0[ε−1] = R; clearly we

haveεm ·R0 ⊂ R′0 ⊂ ε−m · R0 for m ∈ N sufficiently large. It follows thatεm/pn· Φ−nR (R0) ⊂

Φ−nR (R′0) ⊂ ε−m/pn· Φ−nR (R0) for everyn ∈ N. The claim readily follows.

Claim3.4.30. Ra1 is an unramifiedV a-algebra.

Proof of the claim:Let e ∈ R ⊗V [ε−1] R be the idempotent provided by proposition 3.1.4; form ∈ N large enough,εm · e is contained in the subringR0 ⊗V R0. Hence, for everyn ∈ N,εm/p

n· e ∈ Φ−nR (R0)⊗V Φ−nR (R0), soe defines an almost element in(R1⊗V R1)

a which fulfillsthe conditions (i)-(iii) of proposition 3.1.4, and the claim follows.

Claim3.4.31. Ra1 is a uniformly almost finiteV a-algebra.

Proof of the claim:For large enoughm ∈ N we have:R0 ⊂ Φ−1R (R0) ⊂ ε−m · R0, thereforeΦ−nR (R0) ⊂ Φ

−(n+1)R (R0) ⊂ ε−m/p

n· Φ−nR (R0) for everyn ∈ N. By an easy induction we

deduce:Φ−(n+k)R (R0) ⊂∏k

j=0 ε−m/pn+j

· Φ−nR (R0) ⊂ ε−m/pn−1· Φ−nR (R0) for everyn, k ∈ N.

Finally, this implies thatR1 ⊂ ε−m/pn−1· Φ−nR (R0) for everyn ∈ N and the claim follows.

Claim3.4.32. Let S be the integral closure ofV in R; thenRa1 = Sa.

Proof of the claim:Let us endowR with the unique ring topologyτ such that the inducedsubspace topology onR0 is theε-adic topology andR0 is open inR. It is easy to check thatSconsists of power-bounded elements ofR relative to the topologyτ . Since clearlyR1 ⊂ S, itsuffices therefore to show that(Ra

1)∗ ⊂ R is the subring of all power-bounded elements ofR.However,(Ra

1)∗ can be characterized as the subring of allx ∈ R such thatm · x ⊂ R1; thisalready implies that(Ra

1)∗ consists of power-bounded elements. On the other hand, ifx ∈ Ris power-bounded, it follows thatδ · x is topologically nilpotent for everyδ ∈ m; sinceR0 isopen inR, it follows that, for everyδ ∈ m there existsn0 ∈ N such that(δ · x)n ∈ R0 for everyn > n0. By takingn := pk for sufficiently largek ∈ N, we deduce thatΦkR(δ · x) ∈ R0, that isδ · x ∈ R1, and the claim follows.

Claim3.4.33. Ra1 is an almost projectiveV a-algebra.

Proof of the claim:As a special case of claim 3.4.32, letW be the integral closure ofV inV [ε−1]; then:

W a = V a.(3.4.34)

Next, letTrR/V [ε−1] : R → V [ε−1] be the trace map of the finite etale extensionV [ε−1] → R;recall thatTrR/V [ε−1] sends elements integral overV to elements integral overV (to see this, wecan assume thatR has constant rankn overV [ε−1]; then the assertion can be checked after afaithfully flat base changeV [ε−1]→ S, so we can further suppose thatR ≃ V [ε−1]n, in whichcase everything is clear); it then follows from claim 3.4.32and (3.4.34) thatTraR/V [ε−1] restrictsto a morphismT : Ra

1 → V a. Furthermore, lete ∈ R ⊗V [ε−1] R be the idempotent definingthe diagonal imbedding; by claim 3.4.30, for everyδ ∈ m we can writeδ · e =

∑ni xi ⊗ yi for

certainxi, yi ∈ R1. By remark 4.1.17 (whose proof does not use theorem 3.4.28) we deducethe identity:δ · b =

∑ni xi · T (b · yi) for everyb ∈ (Ra

1)∗. This allows us to define morphismsα : Ra

1 → (V a)n, β : (V a)n → Ra1 with β α = δ ·1Ra

1, namelyα(b) = (T (b · y1), ..., T (b · yn))

andβ(v1, ..., vn) =∑n

i xi · vi for everyb ∈ Ra1 andv1, ..., vn ∈ V a

∗ . By lemma 2.4.10, the claimfollows.

Claim3.4.35. The functor (3.4.27) is fully faithful.


Proof of the claim:First of all, it is clear that, for every flatV a-algebrasA, B, the natural map

HomV a-Alg(A,B)→ HomV [ε−1]-Alg(A∗[ε−1], B∗[ε

−1])(3.4.36)

is injective, sinceA∗ ⊂ A∗[ε−1] and similarly forB. Suppose now thatA andB are etale and

almost finite overV a; thenΦA andΦB are automorphisms, due to theorem 3.4.13(ii) and theassumption thatV is perfect. Letψ : A∗[ε

−1]→ B∗[ε−1] be any map ofV [ε−1]-algebras; since

A is almost finite, we haveψ(A∗) ⊂ ε−m · B∗ for m ∈ N large enough. Since Frobenius com-mutes with every ring homomorphism, we deduceψ(A∗) = ψ(Φ−nA∗

(A∗)) ⊂ ε−m/pn·Φ−nB∗

(B∗) =

ε−m/pn·B∗ for everyn ∈ N, soψ induces a morphismψa : A→ B, which shows that (3.4.36)

is surjective.It now follows from claims 3.4.29, 3.4.30, 3.4.31, 3.4.33 that the assignmentR → Ra

1 de-fines an essential inverse to 3.4.27; together with claim 3.4.35, this concludes the proof of thetheorem.


4. FINE STUDY OF ALMOST PROJECTIVE MODULES

4.1. Almost traces. LetA be aV a-algebra.

Definition 4.1.1. Let P be an almost finitely generated projectiveA-module. ThenωP/A is anisomorphism by lemma 2.4.24(b). Thetrace morphismof P is theA-linear morphism

trP/A := evP/A ω−1P/A : EndA(P )

a → A.

We letζP be the unique almost element ofP ⊗A P ∗ such thatωP/A(ζP ) = 1P .

Lemma 4.1.2. LetM , N be almost finitely generated projectiveA-modules, andφ : M → N ,ψ : N → M twoA-linear morphisms. Then :

(i) trM/A(ψ φ) = trN/A(φ ψ).(ii) If ψ φ = a · 1M andφ ψ = a · 1N for somea ∈ A∗, and if, furthermore, there exist

u ∈ EndA(M), v ∈ EndA(N) such thatv φ = φu, thena · (trM/A(u)− trN/A(v)) = 0.

Proof. (i) : by lemma 2.4.24(i), the natural morphismN⊗AalHomA(M,A)→ alHomA(M,N)is an isomorphism (and similarly when we exchange the roles of M andN). ByA-linearity, wecan therefore assume thatφ (resp.ψ) is of the formx 7→ n ·α(x) for somen ∈ N∗, α :M → A(resp. of the formx 7→ m · β(x) for somem ∈ M∗, β : N → A). Then a simple computationyields:

φ ψ = ωN/A(n · α(m)⊗ β) ψ ψ = ωM/A(m · β(n)⊗ α)

and the claim follows directly from the definition of the trace morphism. For (ii) we computeusing (i) :a·trM/A(u) = trM/A(ψφu) = trM/A(ψvφ) = trN/A(vφψ) = a·trN/A(v).

Lemma 4.1.3. LetM ,N be two almost finitely generated projectiveA-modules,φ ∈ EndA(M)andψ ∈ EndA(N). ThentrM⊗AN/A(φ⊗ ψ) = trM/A(φ) · trN/A(ψ).

Proof. As usual we can suppose thatφ = ωM/A(m⊗ α), ψ = ωN/A(n⊗ β) for someα ∈ M∗,β ∈ N∗. Thenφ ⊗ ψ = ωM⊗AN/A((m ⊗ n) ⊗ (α ⊗ β)) and the sought identity follows byexplicit calculation.

Proposition 4.1.4. Let M = (0 → M1i→ M2

p→ M3 → 0) be an exact sequence of al-

most finitely generated projectiveA-modules, and letu = (u1, u2, u3) : M → M be anendomorphism ofM , given by endomorphismsui : Mi → Mi (i = 1, 2, 3). Then we havetrM2/A(u2) = trM1/A(u1) + trM3/A(u3).

Proof. Suppose first that there exists a splittings :M3 → M2 for p, so that we can viewu2 as a

matrix

(u1 v0 u3

), wherev ∈ HomA(M3,M1). By additivity of the trace, we are then reduced

to show thattrM2/A(i v p) = 0. By lemma 4.1.2(i), this is the same astrM3/A(p i v),which obviously vanishes. In general, for anya ∈ m we consider the morphismµa = a · 1M3

and the pull back morphismM ∗ µa →M :

0 // M1i // M2

p // M3// 0

0 // M1// P

p′ //

φ

OO

M3//

µa

OO

0.

Pick a morphismj : M3 → M2 such thatp j = a · 1M3; the pair(j, 1M3) determines amorphismσ : M3 → P such thatσ p′ = 1M3, i.e. the sequenceM ∗ µa is split exact; thissequence also inherits the endomorphismu ∗ µa = (u1, v, u3), for a certainv ∈ EndA(P ).The pair of morphisms(a · 1M2, p) determines a morphismψ : M2 → P , and it is easy to


check thatφ ψ = a · 1M2 andψ φ = a · 1P . We can therefore apply lemma 4.1.2 todeduce thata · (trP/A(v) − trM/A(u2)) = 0. By the foregoing we know thattrP/A(v) =trM1/A(u1) + trM3/A(u3), so the claim follows.

Lemma 4.1.5. LetA be aV a-algebra.

(i) If P :=M ⊗A N is an almost projective and faithful (resp. and almost finitely generated)A-module, than so areM andN .

(ii) If M ⊗A N ≃ A, then the evaluation mapevM :M ⊗AM∗ → A is an isomorphism.(iii) An invertibleA-module is faithful and almost finitely generated projective.(iv) An epimorphismφ :M → N of invertibleA-modules is an isomorphism.

Proof. Clearly (iii) is just a special case of (i). We show (i): by proposition 2.4.23(iv) we knowthatEP/A = A; however, one checks easily thatEP/A ⊂ EN/A, whence

EN/A = A.(4.1.6)

ThereforeN will be faithful, as soon as it is shown to be almost projective, again by virtueof proposition 2.4.23(iv). In any case, (4.1.6) means that,for everyε ∈ m, we can find analmost element of the form

∑ni=1 xi ⊗ φi ∈ N ⊗A N∗, such that

∑ni=1 φi(xi) = ε. We use

such an element to define morphismsA → Nn → A whose composition equalsε · 1N . Aftertensoring byM , we obtain morphismsM → P →M whose composition isε ·1M . Then, sinceP is almost projective, it follows easily that so must beM ; similarly, if P is almost finitelygenerated, the same follows forM . By symmetry, the same holds forN .

(ii): notice that, by (i), we know already thatM andN are almost finitely generated projec-tive. By lemma 4.1.3 we deduce thattrM/A(1M) · trN/A(1N) = 1, so both factors are invertibleinA∗. It follows that the morphismA→ EndA(M) given bya 7→ a ·1M provides a splitting forthe trace morphism (and similarly forN in place ofM). Thus we can writeEndA(M) ≃ A⊕X,EndA(N) ≃ A ⊕ Y for someA-modulesX, Y . However, on the one hand we have a natu-ral isomorphismEndA(M) ⊗A EndA(N) ≃ A; on the other hand, we have a decompositionEndA(M)⊗A EndA(N) ≃ A⊕X ⊕ Y ⊕ (X ⊗A Y ); working out the identifications, one seesthat the induced isomorphismA⊕X ⊕ Y ⊕ (X ⊗A Y ) ≃ A restricts to the identity morphismon the direct summandA; it follows thatX = Y = 0, which readily implies the claim.

(iv): in view of (ii) we can replaceφ by φ ⊗A 1M∗, and thereby assume thatM = A. ThenN ≃ A/Ker(φ); it is clear that such a module is faithful if and only ifKer(φ) = 0. By (iii), theclaim follows.

Lemma 4.1.5 explain why we do not insist, in the definition of an invertibleA-module, thatit should be almost projective or almost finitely generated:both of these conditions can bededuced.

4.1.4.1.7. Suppose now thatB is an almost finite projectiveA-algebra. For anyb ∈ B∗,denote byµb : B → B theB-linear morphismb′ 7→ b · b′. The mapb 7→ µb defines aB-linearmonomorphismµ : B → EndA(B)a. The composition

TrB/A := trB/A µ : B → A

will also be called the almost trace morphism of theA-algebraB.

Proposition 4.1.8. LetA andB be as in(4.1.4.1.7).

(i) If φ : A→ B is an isomorphism, thenTrB/A = φ−1.(ii) If C any otherA-algebra, thenTrC⊗AB/C = 1C ⊗A TrB/A.

(iii) If C is an almost finite projectiveB-algebra, thenTrC/A = TrB/A TrC/B.


Proof. (i) and (ii) are left as exercises for the reader. We verify (iii). It comes down to checkingthat the following diagram commutes:

C ⊗B alHomB(C,B) //

evC/B

C ⊗B alHomA(C,B)∼ // C ⊗A alHomA(C,A)

evC/A

B

TrB/A // A.

Therefore, pickc ∈ C∗ andφ ∈ HomB(C,B). For everyε ∈ m we can find elementsb1, ..., bk ∈B∗ andφ1, ..., φk ∈ HomA(C,A) such thatε · φ(x) =

∑i bi · φi(x) for everyx ∈ C∗. TheB-

linearity ofφ translates into the identity:∑

i

bi · φi(b · x) =∑

i

b · bi · φi(x) for all b ∈ B∗, x ∈ C∗.(4.1.9)

Thenε · evC/B(c⊗ φ) =∑

i bi · φi(c) and we need to show that

TrB/A(∑

i

bi · φi(c)) =∑

i

φi(c · bi).(4.1.10)

For everyi ≤ k, let µi : A → B be the morphisma 7→ bi · a (for all a ∈ A∗); furthermore, letµc : B → C be the morphismb 7→ c · b (for all b ∈ B∗). In view of (4.1.9), the left-hand side of(4.1.10) is equal totrB/A(

∑i µi φi µc). By lemma 4.1.2(i), we havetrB/A(µi φi µc) =

trA/A(φi µc µi) = φi(c · bi) for everyi ≤ k. The claim follows.

Corollary 4.1.11. LetA→ B be a faithfully flat almost finitely presented andetale morphismof almostV -algebras. ThenTrB/A : B → A is an epimorphism.

Proof. Under the stated hypotheses,B is an almost projectiveA-module (by proposition 2.4.13).Let C = Coker(TrB/A) andTrB/B⊗AB the trace morphism for the morphism of almostV -algebrasµB/A. By faithful flatness, the natural morphismC → C ⊗A B = Coker(TrB⊗AB/B)is a monomorphism, hence it suffices to show thatTrB⊗AB/B is an epimorphism (hereB ⊗A Bis considered as aB-algebra via the second factor). However, from proposition4.1.8(i) and (iii)we see thatTrB/B⊗AB is a right inverse forTrB⊗AB/B. The claim follows.

4.1.4.1.12. It is useful to introduce theA-linear morphism

tB/A := TrB/A µB/A : B ⊗A B → A.

We can viewtB/A as a bilinear form; it induces anA-linear morphism

τB/A : B → B∗ = alHomA(B,A)

characterized by the equalitytB/A(b1⊗ b2) = τB/A(b1)(b2) for all b1, b2 ∈ B∗. We say thattB/Ais a perfect pairingif τB/A is an isomorphism.

Lemma 4.1.13.Let A → B be an almost finite projective morphism ofV a-algebras andCanyA-algebra. Denote byηB,C : C ⊗A alHomA(B,A) → alHomC(C ⊗A B,C) the naturalisomorphism provided by lemma 2.4.26(i). Then :

(i) τB/A isB-linear (for the naturalB-module structure ofB∗ defined in remark 2.4.20);(ii) ηB,C isC ⊗A B-linear;(iii) ηB,C (1C ⊗ τB/A) = τC⊗AB/C .

Proof. For anyb ∈ B∗, let ξb : B → A theA-linear morphism defined by the ruleb′ 7→TrB/A(b

′ · b) for all b′ ∈ B∗. Then, directly from the definition we can compute:(ηB,C (1C ⊗τB/A))(c⊗ b)(c

′⊗ b′) = ηB,C(c⊗ ξb)(c′⊗ b′) = c · c′ ·TrB/A(b′ · b) for all b, b′ ∈ B∗, c, c′ ∈ C∗.

But by proposition 4.1.8(ii), the latter expression can be rewritten asτC⊗AB/C(c ⊗ b)(c′ ⊗ b′),

which shows (iii). The proofs of (i) and (ii) are similar direct verifications: we show (i) and


leave (ii) to the reader. Let us pick anyb, b′, b′′ ∈ B∗; then(b · τB/A(b′))(b′′) = τB/A(b′)(bb′′) =

TrB/A(bb′b′′) = (τB/A(bb

′))(b′′).

Theorem 4.1.14.An almost finite projective morphismφ : A → B of almostV -algebras isetale if and only if the trace formtB/A is a perfect pairing.

Proof. By lemma 4.1.13, we have a commutative diagram:

(B ⊗A B)⊗B B∼ //

1B⊗AB⊗BτB

B ⊗A B

1B⊗AτB

τB⊗AB/B

))RRRRRRRRRRRRRR

(B ⊗A B)⊗B B∗∼ // B ⊗A B

∗ηB,B // alHomB(B ⊗A B,B)

(4.1.15)

in which all the morphisms areB⊗AB-linear (here we take theB-module structure onB⊗ABgiven by multiplication on the right factor). Suppose now that φ is etale; then, by corollary3.1.9, there is an isomorphism ofB-algebras:B ⊗A B ≃ IB/A ⊕B. It follows thatτB⊗AB/B =τB/B ⊕ τIB/A/B. Especially,1B ⊗B⊗AB τB⊗AB/B is the identity morphism ofB (by proposition4.1.8(i)). This means that in the diagramB⊗B⊗AB (4.1.15) all the arrows are isomorphisms. Inparticular,τB/A is an isomorphism, as claimed.

To prove the converse, we consider the almost elementζB of theB ⊗A B-moduleB ⊗AB∗. Viewing B∗ as aB-module in the natural way (cp. remark 2.4.20), we also get a scalarmultiplication morphismσB∗/B : B ⊗A B

∗ → B∗ (see (2.2.2.2.5)).

Claim 4.1.16. With the above notation we have:IB/A · ζB = 0 andσB∗/B(ζB) = TrB/A.

Proof of the claim:Notice thatωB/A is alsoB ⊗A B-linear for theB ⊗A B-module structureon EndA(B) such that((b ⊗ b′) · φ)(b′′) = b′ · φ(b · b”) for every b, b′, b” ∈ B∗ and everyφ ∈ EndA(B). We computeωB/A((b ⊗ b′) · ζB)(b”) = ((b ⊗ b′) · ωB/A(ζB))(b”) = b · b′ · b”.Whencex · ζB = µB/A(x) · ζB for everyx ∈ B ⊗A B∗ which implies the first claimed identity.Next we compute:σB∗/B(ζB)(b) = evB((1 ⊗ b) · ζB) = (trB/A ωB/A)((1 ⊗ b) · ζB) =trB/A((1 ⊗ b) · ωB/A(ζB)) = trB/A((1 ⊗ b) · 1B) = TrB/A(b) for everyb ∈ B∗. The claimfollows.

Suppose now thatτB/A is an isomorphism. Then we can definee := (1B ⊗ τ−1B/A)(ζB). From

claim 4.1.16 and lemma 4.1.13(i) we derive thatIB/A · e = 0 andτB/A(σB/B(e)) = TrB/A.The latter equality implies thatσB/B(e) = 1, in other wordsµB/A(e) = 1. We see thereforethate satisfies conditions (ii) and (iii) of proposition 3.1.4 andtherefore also condition (i), sincethe latter is an easy consequence of the other two. ThusA → B is an etale morphism, asclaimed.

Remark 4.1.17. By inspection of the proof of theorem 4.1.14, we see that the following hasbeen shown. LetA→ B be an etale morphism ofV a-algebras. Then(1B ⊗ τB/A)(eB/A) = ζB.

Definition 4.1.18. Thenilradical of an almost algebraA is the idealnil(A) = nil(A∗)a (where,

for a ringR, we denote bynil(R) the ideal of nilpotent elements inR). We say thatA is reducedif nil(A) ≃ 0.

4.1.4.1.19. Notice that, ifR is aV -algebra, then every nilpotent ideal inRa is of the formIa,whereI is a nilpotent ideal inR (indeed, it is of the formIa whereI is an ideal, andm · I isseen to be nilpotent). It follows easily thatnil(A) is the colimit of the nilpotent ideals inA;moreovernil(R)a = nil(Ra). Using this one sees thatA/nil(A) is reduced.

Proposition 4.1.20.LetA → B be anetale almost finitely presented morphism of almost al-gebras. IfA is reduced thenB is reduced as well.


Proof. Under the stated hypothesis,B is an almost projectiveA-module (by virtue of proposi-tion 2.4.13(ii)). Hence, for givenε ∈ m, pick a sequence of morphismsB

uε→ Anvε→ B such

thatvε uε = ε · 1B; let µb : B → B be multiplication byb ∈ B∗ and defineνb : An → An

by νb = vε µb uε. One verifies easily thatνmb = εm−1 · νbm for all integersm > 0. Now,suppose thatb ∈ nil(B∗). It follows thatbm = 0 for m sufficiently large, henceνmb = 0 for msufficiently large. Letp be any prime ideal ofA∗; let π : A∗ → A∗/p be the natural projectionandF the fraction field ofA∗/p. TheF -linear morphismνb∗ ⊗A∗ 1F is nilpotent on the vectorspaceF n, henceπ trAn

∗/A∗(νb∗) = trFn/F (νb∗ ⊗A∗ 1F ) = 0. This shows thattrAn∗/A∗(νb∗)

lies in the intersection of all prime ideals ofA∗, hence it is nilpotent. Since by hypothesisA isreduced, we gettrAn

∗/A∗(νb∗) = 0, whencetrAn/A(νb) = 0. Using lemma 4.1.2(i) we deduceε · trB/A(µb) = 0, and finally,trB/A(b) = 0. Now, for anyb′ ∈ B∗, the almost elementbb′ willbe nilpotent as well, so the same conclusion applies to it. This shows thatτB/A(b) = 0. But byhypothesisB is etale overA, hence theorem 4.1.14 yieldsb = 0, as required.

Remark 4.1.21. LetM be anA-module. We say that an almost elementa of A isM-regularif the multiplication morphismm 7→ am : M → M is a monomorphism. Assume (A) (see(2.1.2.1.6)) and suppose furthermore thatm is generated by a multiplicative systemS which isa cofiltered semigroup under the preorder structure(S ,≻) induced by the divisibility relationin V . We say thatS is archimedeanif, for all s, t ∈ S there existsn > 0 such thatsn ≻ t.Suppose thatS is archimedean and thatA is a reduced almost algebra. ThenS consists ofA-regular elements. Indeed, by hypothesisnil(A∗)

a = 0; since the annihilator ofS in A∗ is 0we getnil(A∗) = 0. Suppose thats ·a = 0 for somes ∈ S anda ∈ A∗. Let t ∈ S be arbitraryand pickn > 0 such thattn ≻ s. Then(ta)n = 0 henceta = 0 for all t ∈ S , hencea = 0.

Definition 4.1.22. Let φ : A→ B be an almost finite projective morphism ofV a-algebras. By(4.1.4.1.12), we can assign toφ aB-linear trace morphismτB/A : B → B∗. Thedifferent idealof the morphismφ is the idealDB/A := AnnB(Coker(τB/A)) ⊂ B.

Lemma 4.1.23.LetM1φ→M2

ψ→ M3 be twoA-linear morphisms of invertibleA-modulesMi

(i ≤ 3) andC anA-algebra. Then:

(i) AnnA(Coker(ψ φ)) = AnnA(Coker(φ)) · AnnA(Coker(ψ)).(ii) AnnC(Coker(1C ⊗A ψ)) = C · AnnA(Coker(ψ)).

Proof. (i): Since, by lemma 4.1.5(iii),M3 is faithfully flat, we have

AnnA(Coker(ψ)) = AnnA(Coker(ψ ⊗A 1M∗3)(4.1.24)

and likewise forφ; hence we can replaceMi byMi⊗AM∗3 and suppose thatM3 = A. Moreover,sinceM2 is invertible,evM2/A is an isomorphism, by lemma 4.1.5(ii). LetevM2/A : M∗2 ⊗AM2 → A be the map given by the rule:φ ⊗ x 7→ evM2/A(x ⊗ φ), for everyφ ∈ (M∗2 )∗ andx ∈M2. Setλ := evM2/A(φ⊗A1M∗

2) :M1⊗AM∗2 → A; thenφ(1M1⊗A evM2/A) = λ⊗A1M2,

so thatAnnA(φ) = AnnA(λ ⊗A 1M2) = AnnA(λ). Thus, we can replaceφ by λ ⊗A 1M2 andthen we have to show that

AnnA(Coker(ψ (λ⊗A 1M2))) = AnnA(Coker(ψ)) · AnnA(Coker(λ)).

However, quite generally we have:

AnnA(Coker(M → A)) = Im(M → A)(4.1.25)

for anyA-linear morphismM → A. Hence we compute:AnnA(Coker(ψ (λ ⊗A 1M2))) =Im(ψ(λ⊗A1M2)) = ψ(Im(λ⊗A1M2)) = ψ(Im(λ)·M2) = Im(λ)·Im(ψ) = AnnA(Coker(λ))·AnnA(Coker(ψ)).

(ii): again, using (4.1.24) we reduce to the case whereM3 = A; then the claim follows easilyfrom (4.1.25).


Lemma 4.1.26.Letφ : A→ B be a morphism ofV a-algebras as in definition(4.1.22). LetCbe anA-algebra. Suppose that eitherC is flat overA, or B∗ is an invertibleB-module for itsnaturalB-module structure. ThenDC⊗AB/C = DB/A · (C ⊗A B).

Proof. Under the stated assumptions,alHomA(B,A) is an almost finitely generated projectiveA-module. In particular,Coker(τB/A) is almost finitely generated; IfC is flat overA, it followsthatAnnC⊗AB(C ⊗A Coker(τB/A)) = DB/A · (C ⊗A B); if B∗ is an invertibleB-module, thesame holds by virtue of lemma 4.1.23(ii). However, by lemma 4.1.13(iii), the trace pairingis preserved under arbitrary base changes, so:C ⊗A Coker(τB/A) ≃ Coker(C ⊗A τB/A) ≃Coker(τC⊗A/B), which shows the claim.

Proposition 4.1.27.Let B → C be a morphism ofA-algebras, and suppose thatB (resp.C) is an almost finite projectiveA-algebra (resp.B-algebra). Suppose moreover thatB∗ :=alHomA(B,A) (resp.C∗ := alHomB(C,B)) is an invertibleB-module (resp.C-module) forits naturalB-module (resp.C-module) structure. Then

DC/A = DC/B ·DB/A.

Proof. LetC∗/A := alHomA(C,A) and define aC-linear morphismξ : alHomB(C,B∗)→ C∗/A

by the rule:φ 7→ (c 7→ φ(c)(1)) for everyφ ∈ HomB(C,B∗) andc ∈ C∗.

Claim 4.1.28. C∗/A is an invertibleC-module andξ is an isomorphism.

Proof of the claim:By lemma 2.4.24(i), the natural morphismλ : C∗⊗BB∗ → alHomB(C,B∗)

is aC-linear isomorphism. It suffices therefore to show thatξ λ−1 : C∗ ⊗B B∗ → C∗/A is an

isomorphism. One verifies easily thatξ λ−1 is defined by the rule:φ ⊗ ψ 7→ ψ φ, and thenthe claim follows from lemma 2.4.26(iii).

Unwinding the definitions, one verifies that the following diagram commutes:

CτC/A //

τC/B

C∗/A

C∗alHomB(C,τB/A)

// alHomB(C,B∗).

ξ

OO

Thus, taking into account claim 4.1.28, and lemma 4.1.23(i), we haveAnnC(Coker(τC/A)) =AnnC(Coker(τC/B)) ·AnnC(Coker(HomB(C, τB/A))). However,Coker(alHomB(C, τB/A)) ≃C∗⊗Coker(τB/A) by lemma 2.4.24(b). By lemma 4.1.5(iii),C∗ is faithfully flat; consequently:

AnnC(Coker(alHomB(C, τB/A))) = AnnC(Coker(τB/A))

which implies the assertion.

Lemma 4.1.29.Letφ : A → B be a morphism ofV a-algebras as in definition(4.1.22). Sup-pose moreover thatB∗ is an invertibleB-module for its naturalB-module structure. Thenφ isetale if and only ifDB/A = B.

Proof. By theorem 4.1.14 it follows easily thatDB/A = B wheneverφ is etale. Conversely,suppose thatDB/A = B; it then follows thatτB/A is an epimorphism. Again by theorem 4.1.14,we need only show thatτB/A is an isomorphism. This follows from lemma 4.1.5(iv).

The following lemma will be useful when we will compute the different ideal in situationssuch as those contemplated in proposition 5.3.11.

Lemma 4.1.30.LetA be aV a-algebra,B an almost finite almost projectiveA-algebra, andlet Bα | α ∈ J be a net ofA-subalgebras ofB, withBα almost finite projective overA foreveryα ∈ J , such thatlim

α∈JBα = B in IA(B). Thenlim

α∈JDBα/A = DB/A.


Proof. For givenα ∈ J , let ε ∈ V such thatε · B ⊂ Bα; lemma 4.1.2(ii) implies thatε ·TrBα/A(b) = ε · TrB/A(b) for everyb ∈ Bα∗. Hence the diagrams:

Bµε //

ε·τB/A

Bα

τBα/A

Bα

τBα/A

// B

ε·τB/A

B∗ // B∗α B∗α

µ∗ε // B∗

commute. The rightmost diagram implies thatDBα/A · Im(µ∗ε) ⊂ Im(ε · τB/A) ⊂ Im(τB/A).Henceε ·DBα/A ⊂ DB/A, so finallyAnnV (B/Bα) ·DBα/A ⊂ DB/A. From the leftmost diagramwe deduce thatε · DB/A (which is an ideal inBα) annihilatesCoker(ε · τB/A : B → B∗)and on the other handAnnV (B/Bα) obviously annihilatesCoker(B∗ → B∗α); we deduce thatAnnV (B/Bα)

2 ·DB/A ⊂ DBα/A, whence the claim.

4.2. Endomorphisms of Gm. This section is dedicated to a discussion of the universal ringthat classifies endomorphisms of the formal groupGm. The results of this section will be usedin sections 4.3 and 4.4.

4.2.4.2.1. For every ringR and every integern ≥ 0 we introduce the ”n-truncated” version ofGm,R. This is the schemeGm,R(n) := Spec(R[T ]/(T n+1)), endowed with the multiplicationmorphism which is associated to the co-multiplication map

R[T ]/(T n+1)→ R[T, S]/(T, S)n+1 T 7→ T + S + T · S.

Then in the category of formal schemes we have a natural identificationGm,R ≃ colimn∈N

Gm,R(n).

4.2.4.2.2. In the terminology of [39,§II.4], Gm(n) is then-bud ofGm. We will be mainlyinterested in the endomorphisms ofGm(n), but before we can get to that, we will need somecomplements on buds over artinian ring. Therefore, supposewe have a cartesian diagram ofartinian rings

R3//

R1

R2

// R0

(4.2.3)

such that one of the two mapsRi → R0 (i = 1, 2) is surjective. For any ringR, we definethe categoryBud(n, d, R) of n-buds overR whose underlyingR-algebra is isomorphic toR[T1, ..., Td]/(T1, ..., Td)

n+1.

Lemma 4.2.4. LetS be a finite flat augmented algebra over a local noetherian ringR; let I bethe augmentation ideal, and suppose thatIn+1 = 0. Letκ be the residue field ofR, and supposethatS ⊗R κ ≃ κ[t1, ..., td]/(t1, ..., td)

n+1. ThenS ≃ R[T1, ..., Td]/(T1, ..., Td)n+1.

Proof. Let ε : S → R be the augmentation map. For everyi = 1, ..., d, pick a liftingT ′i ∈ S ofti; setTi = T ′i − ε(T

′i ). By Nakayama’s lemma, the monomialsT a11 · ... · T

add with

∑di=1 ai ≤ n

generate theR-moduleS. Furthermore, under the stated hypothesis,S is a freeR-module, andits rank is equal todimκ S⊗R κ; hence the above monomials form anR-basis ofS. Clearly theelementsTi lie in the augmentation ideal ofS, therefore every product ofn + 1 of them equalszero; in other words, the natural morphismR[X1, ..., Xd] → S given byXi 7→ Ti is surjective,with kernel containingJ := (X1, ..., Xd)

n+1; but by comparing the ranks overR we see thatthis kernel cannot be larger thanJ . The assertion follows.


Proposition 4.2.5. In the situation of(4.2.3), the natural functor

Bud(n, d, R3)→ Bud(n, d, R1)×Bud(n,d,R0) Bud(n, d, R2)

is an equivalence of categories.

Proof. Let Mi,proj (i = 0, ..., 3) be the category of projectiveRi-modules. By our previousdiscussion on descent, we already know that (4.2.3) inducesa natural equivalence betweenM3,proj and the2-fibered productM1,proj×M0,proj

M2,proj. It is easy to see that this equivalencerespects the rank ofRi-modules, hence induces a similar equivalence for the categoriesMi,f.f. offreeRi-modules of finite rank. Given two objectsM := (M1,M2, α :M1⊗R1R0

∼→M2⊗R2R0)

andN := (N1, N2, β : N1 ⊗R1 R0∼→ N2 ⊗R2 R0), define the tensor productM ⊗ N :=

(M1⊗R1 N1,M2⊗R2 N2, α⊗R0 β). Then one checks easily that the above equivalences respecttensor products. It follows formally that one has analogousequivalences for the categories offinite flatRi-algebras. From there, one further obtains equivalences onthe categories of suchRi-algebras that are augmented overRi, and even on the subcategoriesRi-Alg

(n)aug.fl. of those

augmentedRi-algebras such that the(n+1)-th power of the augmentation ideal vanishes. Thesecategories admit finite coproducts, that are constructed asfollows. For augmentedRi-algebrasεA : A → Ri andεB : B → Ri, set(A → Ri) ⊗ (B → Ri) := (A ⊗Ri

B/Ker(εA ⊗Ri

εB)n+1 → Ri); this is a coproduct ofA andB. By formal reasons, the foregoing equivalences

of categories respect these coproducts. Finally, an objectof Bud(n, d, Ri) can be defined as acommutative group object in(Ri-Alg

(n)aug.fl.)

o, such that its underlyingRi-algebra is isomorphictoRi[T1, ..., Td]/(T1, ..., Td)

n+1. By formal categorical considerations we see that the foregoingequivalence induces equivalences on the commutative groupobjects in the respective categories.It remains to check that anR3-algebraS such thatS ⊗R3 Ri ≃ Ri[T1, ..., Td]/(T1, ..., Td)

n+1,(for i = 1, 2) is itself of the formRi[T1, ..., Td]/(T1, ..., Td)

n+1. However, this follows readilyfrom lemma 4.2.4 and the fact that one of the mapsR3 → Ri (i = 1, 2) is surjective.

4.2.4.2.6. For a given ringR, the endomorphisms ofGm,R(n) are all the polynomialsf(T ) :=a0+a1 ·T+...+an ·T

n such thatf(T )+f(S)+f(T )·f(S)≡ f(T+S+T ·S) (mod(T, S)n+1).This relationship translates into a finite set of polynomialidentities for the coefficientsa0, ..., an,and using these identities we can therefore define a quotientGn of the ring inn indeterminatesZ[X1, ..., Xn] which will be the “universal ring of endomorphisms” ofGm(R), i.e., such thatX1 ·T +X2 ·T

2+ ...+Xn ·Tn is an endomorphism ofGm,Gn(n) and such that, for every ringR,

and everyf(T ) as above, the mapZ[X1, ..., Xn] → R given byXi 7→ ai (i = 1, ..., n) factorsthrough a (necessarily unique) mapGn → R. One of the main results of this section will be asimple and explicit description of the ringGn.

Proposition 4.2.7. Gn is a smoothZ-algebra.

Proof. We know already thatGn is of finite type overZ, therefore it suffices to show that, forevery prime idealp of Gn, the local ringGn,p is formally smooth for thep-adic topology (see[21, Ch.IV, Prop.17.5.3]). Therefore, letR1 → R0 be a surjective homomorphism of localartinian rings; we need to show that the natural mapEnd(Gm,R1(n)) → End(Gm,R0(n)) issurjective. Letf ∈ End(Gm,R0(n)); we define an automorphismχ of Gm,R0 ×R0 Gm,R0 :=

R0[T, S]/(T, S)n+1, then-bud of Gm × Gm, by setting(T, S) 7→ (T, f(T ) + S + f(T ) · S).

Then, thanks to proposition 4.2.5, we obtain ann-budXn overR2 := R1 ×R0 R1, by gluingtwo copies ofGm,R1 ×Gm,R1 along the automorphismχ.

Claim 4.2.8. Then-budXn is isomorphic toGm,R2 × Gm,R2 if and only if χ lifts to an auto-morphism ofGm,R1 ×Gm,R1 .

Proof of the claim:Taking into account the decription ofB(n, d, R2) as2-fibered product ofcategories, the proof amounts to a simple formal verification, which is best left to the reader.


Claim4.2.9. There exists a compatible system ofk-budsXk overR2 for everyk > n, such thatXk reduces toXk−1 overR2, and specializes toG2

m,R1(k) over the quotientR1 of R2.

Proof of the claim:In caseR2 is a torsion-freeZ-algebra, this follows from [39, Ch.II,§4.10]and an easy induction. IfR2 is a general artinian ring, choose a torsion-freeZ-algebraR3 with asurjective homomorphismR3 → R2. By loc. cit. (and an easy induction) we can find ann-budYn overR3 such thatYn specialises toXn on the quotientR2, andYn reduces toG2

m,R3(1) over

R3. Then, again byloc. cit, we can find a compatible system ofk-budsYk on R3 for everyk > n, such thatYk reduces toYk−1 overR3 and specializes toG2

m,R1(k) over the quotientR1

of R3. The claim holds if we takeXk equal to the specialization ofYk overR2.

The direct limit (in the category of formal schemes) of the system(Xk)k≥n is a formal groupX overR2, such thatX⊗R2 R1 ≃ Gm,R1 × Gm,R1 . This formal group gives rise to ap-divisiblegroup(X(n))n≥0, whereX(n) is the kernel of multiplication bypn in X. For everym ∈ N,X(m) is a finite flat group scheme overR2, such thatX(m)×R2R1 ≃ µpm,R1×µpm,R1 . Denoteby X(m)∗ the Cartier dual ofX(m) (cp. [43,§III.14]). ThenX(m)∗ ×R2 R1 ≃ (Z/pmZ)2R1

, inparticular it hasp2m connected components. Since the pair(R2, R1) is henselian, it follows thatX(m)∗ must havep2m connected components as well, and consequentlyX(m)∗ ≃ (Z/pmZ)2R2

.

Finally, this shows thatX(n) ≃ µpm,R2 × µpm,R2 , whenceX ≃ Gm,R2 × Gm,R2 . Fromclaim 4.2.8, we deduce thatχ lifts to an automorphismχ′ of Gm,R1(n) × Gm,R1(n). Leti : Gm,R1(n)→ Gm,R1(n)×Gm,R1(n), π : Gm,R1(n)×Gm,R1(n)→ Gm,R1(n) be respectivelythe imbedding of the first factor, and the projection onto thesecond factor; clearlyπ χ′ iyields a lifting off(T ), as required.

4.2.4.2.10. Next, let us remark that, for everyn ∈ N, the polynomial(1 + T )X − 1 :=X · T +

(X2

)· T 2 + ... +

(Xn

)· T n ∈ Q[X, T ] is an endomorphism ofGm,Q[X](n). As a

consequence, there is a unique ring homomorphismGn → Z[X,(X2

), ...,

(Xn

)] representing this

endomorphism. The following theorem will show that this homomorphism is an isomorphism.

Theorem 4.2.11.The functor

Z-Alg→ Set R 7→ EndR(Gm,R(n))

is represented by the ringZ[X,(X2

), ...,

(Xn

)].

Proof. The above discussion has already furnished us with a naturalsurjective mapρ : Gn →Z[X,

(X2

), ...,

(Xn

)]. Therefore, it suffices to show that this map is injective.

Claim4.2.12. ρ⊗Z 1Q is an isomorphism.

Proof of the claim:First of all, the mapρ can be characterized in the following way. The identitymapGn → Gn determines an endomorphismf(T ) := a0 + a1 · T + ... + an · T n of Gm,Gn(n);thenρ is the unique ring homomorphism such thatρ(f) := f(a0)+f(a1) ·T + ...+f(an) ·T n =(1 + T )X − 1. On the other hand, the ringGn ⊗Z Q represents endomorphisms ofGm(n) inthe category ofQ-algebras. However, for everyn ∈ N and for everyQ-algebraR, there is anisomorphism

log : Gm,R(n)∼−→ Ga,R(n)

to then-bud of the additive formal groupGa,R. The endomorphism group ofGa,R(n) is easilycomputed, and found to be isomorphic toR. In other words, the universal ring representing en-domorphisms ofGa(n) overQ-algebras is justQ[X ], and the bijectionHomQ-Alg(Q[X ], R) ≃End(Ga,R(n)) assigns to a homomorphismφ : Q[X ] → R, the endomorphismgφ(T ) :=φ(X) ·T . It follows that, for anyQ-algebraR there is a natural bijectionHomQ-Alg(Q[X ], R) ≃End(Gm,R(n)) given by:(φ : Q[X ]→ R) 7→ exp(φ(X) · log(1 + T ))− 1 = (1 + T )φ(X) − 1.


Especially,f(T ) can be written in the form(1 + T )ψ(X) − 1 for a unique ring homomorphismψ : Q[X ]→ Gn ⊗Z Q. Clearlyψ is inverse toρ⊗Z 1Q.

In view of claim 4.2.12, we are thus reduced to show thatGn is a flatZ-algebra, which followsreadily from proposition 4.2.7.

4.2.4.2.13. Furthermore,Gn is endowed with a co-addition,i.e. a ring homomomorphismGn → Gn ⊗Z Gn satisfying the usual co-associativity and co-commutativity conditions. Theco-addition is given by the rule:

coadd : Gn → Gn ⊗Z Gn

(X

k

)7→

∑

i+j=k

(X

i

)⊗

(X

j

).

Moreover, for everyk ∈ Z, we have a ring homomorphismπk : Gn → Z, which corresponds tothe endomorphism ofGm,Z(n) given by the rule:T 7→ (1 + T )k − 1 (raising to thek-th powerin Gm,Z(n)). Hence we derive, for everyk ∈ Z, a ring homomorphism

Gncoadd // Gn ⊗Z Gn

1Gn⊗πk // Gn.(4.2.14)

Remark 4.2.15. (i) On Gn ⊗Z Q = Q[X ], (4.2.14) is the unique map such that(Xi

)7→(X+ki

)

for all i ≤ n, therefore we see that(X+ki

)∈ Gn for all k ∈ Z, n ≥ 0 and0 ≤ i ≤ n. Moreover,

(4.2.14) is clearly an automorphism for everyk ∈ Z.(ii) It is also interesting (though it will not be needed in this work) to remark thatGn is

endowed additionally with a co-composition structure, so thatGn is actually a co-ring, and itrepresents the functorR 7→ End(Gm,R(n)) from Z-algebras to unitary rings. One can checkthat the co-composition map is given by the rule:

(X

k

)7→∑

φ

(X

φ

)⊗∏

j∈N∗

(Y

j

)φ(j)

whereφ ranges over all the functionsφ : N∗ := N \ 0 → N subject to the condition that∑j∈N∗ j · φ(j) = k, and

(Xφ

):=

X(X−1)·...·(X−∑

j∈N∗ φ(j)+1)∏j∈N∗ φ(j)!

. To show that(Xφ

)∈ Gn, one notices

that(Xφ

)=∏

j∈N∗

(X−

∑j−1i=1 φ(i)

φ(j)

)and then uses (i).

4.2.4.2.16. For the rest of this section we fix a prime numberp and we letvp : Q→ Z∪ ∞be thep-adic valuation.

Lemma 4.2.17.The ringGn,(p) := Gn⊗ZZ(p) is theZ(p)-algebra generated by the polynomialsX,(Xp

),(Xp2

),...,(Xpk

), wherek is the unique integer such thatpk ≤ n < pk+1.

Proof. We proceed by induction onn. It suffices to prove that(Xn

)is contained in theZ(p)-

algebraR := Z(p)[(Xp

),(Xp2

), ...,

(Xpk

)]. We will use the following (easily verified) identity which

holds inQ[X ] for everyi, j ∈ N :(

X

i+ j

)=

(X

i

)·

(X − i

j

)·

(i+ j

j

)−1.(4.2.18)

Suppose first thatn is a multiple ofpk, and writen = (b + 1)pk for someb < p− 1. If b = 0,there is nothing to prove, so we can even assume thatb > 0. We apply (4.2.18) withi = b · pk

andj = pk. By remark 4.2.15(i),(X−b·pk

pk

)is in R, and so is

(Xb·pk

), by induction. The claim

will therefore follow in this case, if we show that((b+1)pk

pk

)is invertible inZ(p). However, this is

clear, sincevp(i) = vp(i + b · pk) for everyi = 1, ..., pk. Finally, it remains consider the case


wheren = b · pk + a for someb > 0 and0 < a < pk. This is dealt with in the same way: apply(4.2.18) withi = b · pk andj = a and use the previous case.

Lemma 4.2.19.Let k ∈ N. If R is a flatZ(p)-algebra andf ∈ R, then the following twoconditions are equivalent:

(i)(fpi

)∈ R for everyi = 1, ..., k;

(ii) locally onSpec(R) there existsj ∈ Z such thatf ≡ j (mod pk).

Proof. We may assume thatR is local. Fork = 0 there is nothing to prove. Fork = 1 we have(fp

)= u ·p−1 ·

∏p−1i=0 (f − i) for a unitu of R. Then the assertion holds since all but possibly one

of thef − i are invertible. Fork > 1, by induction we can writef = i+ p · g for someg ∈ Rand0 ≤ i < p. Sincevp(pk!) = 1 + p+ p2 + ...+ pk−1, we have

(f

pk

)= u · p−1−p−p

2−...−pk−1

·∏

j≡i (mod p)

0≤j<pk

(f − j) = u′ ·

(g

pk−1

)

for some unitsu, u′ ∈ R. The claim follows.

4.2.4.2.20. For every integerk ≥ 0, we construct a schemeXk by gluing the affine schemesSpec(Z(p)[

X−ipk

]) (0 ≤ i < pk) along their general fibres. For everyk ∈ N and everyi ∈ N with

0 ≤ i < pk+1 there is an obvious imbeddingZ(p)[X−ipk

] ⊂ Z(p)[X−ipk+1 ]. By gluing the duals of these

imbeddings, we obtain, for everyk ∈ N, a morphism of schemesρk : Xk+1 → Xk. Let alsoξk : Spec(Gpk+1)→ Spec(Gpk) be the morphism which is dual to the imbeddingGpk ⊂ Gpk+1.

Proposition 4.2.21.With the notation of(4.2.4.2.20)we have:

(i) For givenn > 0, let k be the unique integer such thatpk ≤ n < pk+1. Then there is anatural isomorphism of schemes:πk : Xk

∼−→ Spec(Gn ⊗Z Z(p)).

(ii) For everyk ∈ N the diagram of schemes:

Xk+1

πk+1 //

ρk

Spec(Gpk+1 ⊗Z Z(p))

ξk⊗ZZ(p)

Xkπk // Spec(Gpk ⊗Z Z(p))

commutes.

Proof. By lemma 4.2.17 we may assume thatn = pk. By lemma 4.2.19, we see that bothXk

andSpec(Gpk ⊗Z Z(p)) represent the same functor from the category of flatZ(p)-schemes to thecategory of sets. Since both schemes are flat overSpec(Z(p)), (i) follows. It is similarly clearthatξk⊗ZZ(p) andρk represent the same natural transformation of functors, so (ii) follows.

Corollary 4.2.22. (i) For givenn ∈ N, let k be the unique integer such thatpk−1 ≤ n < pk.Then there is a natural ring isomorphism

Z/pkZ∼−→ End(Gm,Fp(n)) i 7→ (1 + T )i − 1

(ii) LetR be a ring such thatFp ⊂ R. Then there is a natural ring isomorphism

C0(SpecR,Zp)

∼−→ EndR(Gm,R) β 7→ (1 + T )β − 1.

Proof. (i): by lemma 4.2.17 we can assumen = pk − 1. In this case, it is clear that thepolynomials(1 + T )i − 1 are all distinct fori = 0, ..., pk − 1 and they form a subring ofEnd(Gm,Fp(n)). However, an endomorphism ofGm,Fp(n) corresponds to a unique point inSpec(Gn)(Fp). From proposition 4.2.21(i) we derive thatSpec(Gn ⊗Z Fp) is the union of the


special fibres of the affine schemesSpec(Z[X−ipk−1 ]), for i = 0, ..., pk−1 − 1. Each of those

contribute an affine lineA1Fp

, soSpec(Gn⊗ZFp) consists of exactlypk−1 connected components.In total, we have therefore exactlypk points inSpec(Gn)(Fp), so (i) follows.

(ii): to give an endomorphism ofGm is the same as giving a compatible system of endomor-phisms ofGm(n), one for eachn ∈ N. In caseFp ⊂ R, lemma 4.2.17 shows that this is alsoequivalent to the datum of a compatible system of morphismsφk : SpecR→ Spec(Gpk ⊗Z Fp),for everyk ≥ 0. From proposition 4.2.21(ii) we can further deduce that, under the morphismξk, each of thepk+1 connected components ofSpec(Gpk+1 ⊗Z Fp) gets mapped onto one ofthe pk+1 rational points ofSpec(Gpk ⊗Z Fp). Sinceφk−1 = ξk φk, we see that the imageof φk−1 is contained inSpec(Gpk−1)(Fp), for everyk > 0. Taking (i) into account, we seethat an endomorphism ofGm,R is the same as the datum of a compatible system of continuousmapsSpec(R) → Z/pkZ. Since thep-adic topology ofZp is the inverse limit of the discretetopologies on theZ/pkZ, the claim follows.

4.3. Modules of almost finite rank. Let A be aV a-algebra,P an almost finitely generatedprojectiveA-module andφ ∈ EndA(P ).

4.3.4.3.1. We say thatφ is Λ-nilpotent if there exists an integeri > 0 such thatΛiAφ =0. Notice that theΛ-nilpotent endomorphisms ofP form a bilateral ideal of the unitary ringEndA(P ). Notice also thatΛiAP is an almost projectiveA-module for everyi ≥ 0; indeed, thisis easily shown by means of lemma 2.4.10. For aΛ-nilpotent endomorphismφ we introducethe notation

det(1P + φ) :=∑

i≥0

trΛiAP/A

(ΛiAφ).

Notice that the above sum consists of only finitely many non-zero terms, so thatdet(1P + φ) isa well defined element ofA∗.

Lemma 4.3.2. LetP be an almost finitely generated projectiveA-module.

(i) If φ is aΛ-nilpotent endomorphism ofP andα : A→ A′ is any morphism ofV a-algebras,setP ′ := P ⊗A A

′. Then:det(1P ′ + φ⊗A 1A′) = α(det(1P + φ)).(ii) Letφ, ψ ∈ EndA(P ) such thatφ ψ andψ φ areΛ-nilpotent. Then:det(1P + φ ψ) =

det(1P + ψ φ).

Proof. (i) is a straightforward consequence of the definitions. As for (ii), it is clear thatψ φ isΛ-nilpotent and the stated identity follows directly from lemma 4.1.2(i).

4.3.4.3.3. Now, letφ, ψ ∈ EndA(P ) be two endomorphisms. SetB := A[X, Y ]/(Xn, Y n)andPB := P ⊗A B; φ andψ induce endomorphisms ofPB that we denote again by the sameletters. ClearlyX · φ andY · ψ areΛ-nilpotent; hence we get elementsdet(1PB

+ X · φ),det(1PB

+Y ·ψ) anddet(1PB+X ·φ+Y ·ψ+XY ·ψφ) inB∗. Notice that any element ofB∗

can be written uniquely as anA∗-linear combination of the monomialsX iY j with 0 ≤ i, j < n.Moreover, it is clear thatdet(1PB

+X · φ) =∑

0≤i<n trΛiAP/A

(ΛiAφ) ·Xi, and similarly forψ.

Proposition 4.3.4. With the above notation, the following identity holds :

det(1PB+X · φ) · det(1PB

+ Y · ψ) = det(1PB+X · φ+ Y · ψ +XY · ψ φ).(4.3.5)

Proof. First of all we remark that, whenP is a freeA-module of finite rank, the above identityis well-known, and easily verified by working with matrices with entries inA∗. Suppose nextthatP is arbitrary, butφ = ε · φ′, ψ = ε · ψ′ for someφ′, ψ′ ∈ EndA(P ) andε ∈ m. Pick a freeA-moduleF of finite rank, and morphismsu : P → F , v : F → P such thatv u = ε · 1P .Setφε := u φ′ v : F → F and define similarlyψε. Clearlydet(1PB

+X · φ) = det(1PB+

X · ε · φ′) = det(1PB+X · v u φ′) = det(1PB

+X · φε) (by lemma 4.3.2(ii)) and similarly


for the other terms appearing in (4.3.5). Thus we have reduced this case to the case of a freeA-module. Finally, we deal with the general case. The foregoing shows that the sought identityis known at least whenφ andψ are replaced byε · φ, resp.ε · ψ, for anyε ∈ m. Equivalently,consider theA-algebra endomorphismα : B → B defined byX 7→ ε ·X, Y 7→ ε · Y and letCbe theB-algebra structure onB determined byα; by lemma 4.3.2(i) we have

det(1PB+X · ε · φ) = det(1PC

+ (X · φ)⊗B 1C) = α(det(1PB+X · φ))

and similarly for the other terms appearing in (4.3.5). Thus, the images underα of the twomembers of (4.3.5) coincide. But applyingα to a monomial of the forma ·X iY j has the effectof multiplying it by εi+j; by (B), the (i + j)-powers of elements ofm generatem, hence theclaim follows easily.

Corollary 4.3.6. If φ, ψ ∈ EndA(P ) are twoΛ-nilpotent endomorphisms, then

det(1P + φ) · det(1P + ψ) = det(1P + φ+ ψ + φ ψ).

Proof. For an arbitraryα ∈ EndA(P ), one can defineP ′ := P⊗AA[[X ]] anddet(1P ′+X ·α) :=∑i≥0 trΛi

AP/A(ΛiAα) ·X

i ∈ A∗[[X ]]. Then proposition 4.3.4 implies that the analogue of (4.3.5)holds inA∗[[X, Y ]]. But if α is Λ-nilpotent, the power seriesdet(1P ′ + X · α) is actually apolynomial inA∗[X ]; the claim then follows by evaluating the polynomialsdet(1P ′ +X · φ),det(1P ′ + Y · ψ) anddet(1P ′ +X · φ+ Y · ψ +XY · φ ψ) for X = Y = 1.

4.3.4.3.7. Next, forP as above, set

χP (X) :=∑

i≥0 trΛiAP/A

(ΛiA1P ) ·Xi ∈ A∗[[X ]]

ψP (X) :=∑

i≥0 trSymiAP/A

(1SymiAP

) ·X i ∈ A∗[[X ]].

Corollary 4.3.8. LetP be an almost finitely generated projectiveA-module. Then:

(i) the power seriesχP (X) defines an endomorphism of the formal groupGm,A∗.(ii) χP (X) · ψP (−X) = 1.(iii) χP (X) ∈ 1 + EP/A∗[[X ]].

Proof. (i) is immediate. For (ii), recall that, for everyn > 0 there is an acyclic Koszul complex(cp. [12, Ch.X,§9, n.3, Prop.3])

0→ ΛnAQ→ (Λn−1A Q)⊗A (Sym1AQ)→ ...→ (Λ1

AQ)⊗A (Symn−1A Q)→ Symn

AQ→ 0.

From proposition 4.1.4 we derive, by a standard argument, that the trace is an additive functionon arbitrary bounded acyclic complexes. Then, taking into account lemma 4.1.3 we obtain:∑n

i=0(−1)i · trΛn−i

A P/A(1Λn−iA P ) · trSymi

AP/A(1Symi

AP) = 0 for everyn > 0. This is equivalent to

the sought identity. To show (iii) we remark more precisely thatEΛrAP/A

⊂ EP/A for everyr > 0.Indeed, setB := A/EP/A. Then, by proposition 2.4.23(i),(iii):EΛr

AP/A·B = EΛr

B(P⊗AB)/B = 0,whence the claim.

Definition 4.3.9. Let P be an almost finitely generated projectiveA-module.

(i) The formal rankof P is the ring homomorphismf.rkA(P ) : G∞ := Z[α,(α2

), ...] → A∗

associated toχP (X).(ii) We say thatP is of almost finite rankif, for everyε ∈ m, there exists an integeri ≥ 0 such

thatε · ΛiAP = 0.(iii) We say thatP is of finite rankif there exists an integeri ≥ 0 such thatΛiAP = 0.(iv) Let r ∈ N; we say thatP hasconstant rank equal tor if Λr+1

A P = 0 andΛrAP is aninvertibleA-module.


Remark 4.3.10. (i): It follows easily from lemma 2.3.8(vi) that every uniformly almost finitelygenerated projectiveA-module is of finite rank.

(ii): Notice that if P is of finite rank, thenχP (X) is a polynomial, whence it defines anendomorphism of the algebraic groupGm,A∗. In this case, it follows thatχP (X) is of the form(1 + X)α, whereα : Spec(A∗) → Z is a continuous function (whereZ is seen as a discretetopological space). More precisely, there is an obvious injective ring homomorphism

C0(Spec(A∗),Z)→ EndA(Gm,A∗) β 7→ (1 +X)β(4.3.11)

which allows to identify the continuous functionα with the formal rank ofP . Moreover, ifΛiAP = 0, it is clear thatα(Spec(A∗)) ⊂ 0, ..., i− 1.

The main result of this section is theorem 4.3.28, which describes general modules of al-most finite rank as infinite products of modules of finite rank.The first step is lemma 4.3.12,concerned with the case of anA-module of rank one.

Lemma 4.3.12.LetP be an almost finitely generated projectiveA-module such thatΛ2AP = 0.

There existsV a-algebrasA0, A1 and an isomorphism ofV a-algebrasA ≃ A0 × A1 such thatP ⊗A A0 = 0 andP ⊗A A1 is an invertibleA1-module.

Proof. Since the natural mapP × P → Λ2AP is universal for alternatingA-bilinear maps on

P × P , we have

f(p) · q = f(q) · p for everyf ∈ (P ∗)∗ andp, q ∈ P∗.(4.3.13)

Using (4.3.13) we deriveωP/A(p ⊗ f)(q) = trP/A(ωP/A(p ⊗ f)) · q for everyf ∈ (P ∗)∗ andp, q ∈ P∗. In other words,ωP/A(p⊗f) = trP/A(ωP/A(p⊗f)) ·1P , for everyf ∈ (P ∗)∗, p ∈ P∗.By linearity we finally deduce

φ = trP/A(φ) · 1P for all φ ∈ EndA(P ).(4.3.14)

Now, by remark 4.3.10(ii), the hypothesisΛ2AP = 0 also implies thatχP (X) = (1+X)α, for a

continuous functionα : Spec(A∗)→ 0, 1. We can decompose accordinglyA = A0 × A1, sothatα(Spec(Ai∗)) = i, which gives the sought decomposition. We can now treat separately thetwo casesA = A0 andA = A1. In casef.rkA(P ) = 0, thentrP/A(1P ) = 0, and then (4.3.14)implies thatP = 0. In casef.rkA(P ) = 1, thentrP/A(1P ) = 1 and (4.3.14) implies that thenatural mapA→ EndA(P ) : a 7→ a · 1P is an inverse fortrP/A, thusA ≃ P ⊗A P ∗.

The next step consists in analyzing the structure ofA-modules of finite rank. To this purposewe need some preliminaries of multi-linear algebra.

4.3.4.3.15. For everyn ≥ 0 let n := 1, ..., n; for a subsetI ⊂ n let |I| be the cardinality ofI; for a given partitionn = I ∪ J , let≺ denote the total ordering onn that restricts to the usualordering onI and onJ , and such thati ≺ j for everyi ∈ I, j ∈ J . Finally letεIJ be the signof the unique order-preserving bijection(n, <)→ (n,≺).

Let M be anyA-module. Given elementsm1, m2, ..., mn in M∗, andI ⊂ n a subset ofelementsi1 < i2 < ... < i|I|, letmI := mi1 ∧ ... ∧ mi|I| ∈ Λ

|I|AM∗ (with the convention that

m∅ = 1 ∈ A∗ = Λ0AM∗).

4.3.4.3.16. LetM,N be any twoA-modules. For everyi, j ≥ 0, there is a natural morphism

ΛiAM ⊗A ΛjAN → Λi+jA (M ⊕N)(4.3.17)

determined by the rule:

m1 ∧ ... ∧mi ⊗ n1 ∧ ... ∧ nj 7→ (m1, 0) ∧ (m2, 0) ∧ ... ∧ (0, n1) ∧ ... ∧ (0, nj)


for all m1, ..., mi ∈ M∗ andn1, ..., nj ∈ N . The morphisms (4.3.17), assemble to an isomor-phism ofA-modules

Λ•AM ⊗A Λ•AN → Λ•A(M ⊕N).(4.3.18)

Clearly, there is a unique gradedA-algebra structure onΛ•AM ⊗A Λ•AN such that (4.3.18) isan isomorphism of (graded-commutative)A-algebras. Explicitly, givenxi ∈ ΛaiAM , yi ∈ ΛbiAN(i = 1, 2) one verifies easily that the product onΛ•AM ⊗A Λ•AN is fixed by the rule

(x1 ⊗ y1) · (x2 ⊗ y2) = (−1)a2b1 · (x1 ∧ x2)⊗ (y1 ∧ y2).(4.3.19)

ThenΛ•AM ⊗A Λ•AN is even a bigradedA-algebra, if we letΛiAM ⊗A ΛjAN be the gradedcomponent of bidegree(i, j).

4.3.4.3.20. Next, letδ : M → M ⊕ M be the diagonal morphismm 7→ (m,m) (for allm ∈ M∗). It induces a morphismΛ•Aδ : Λ•AM → Λ•A(M ⊕M) of A-algebras. We let∆ :Λ•AM → Λ•AM ⊗A Λ•AN be the composition of the morphismΛ•Aδ and the inverse of theisomorphism (4.3.18). For everya, b ≥ 0 we also let∆a,b : Λ•AM → ΛaAM ⊗A ΛbAN bethe composition of∆ and the projection onto the graded component of bidegree(a, b). Themorphisms∆a,b are usually called ”co-multiplication morphisms”. An easycalculation showsthat:

∆a,b(x1 ∧ x2 ∧ ... ∧ xa+b) =∑

I,J

εIJ · xI ⊗ xJ(4.3.21)

where the sum ranges over all the partitionsa+ b = I ∪ J such that|I| = a. Let nowx1, ..., xa, y1, ..., yb ∈ M∗. Since∆ is a morphism ofA-algebras, we have∆(xa ∧ yb) =∆(xa) ·∆(yb). Hence, using (4.3.19) and (4.3.21) one deduces easily:

∆a,b(xa ∧ yb) =∑

I,J,K,L

εIJ · εKL · (−1)|J | · (xI ∧ yK)⊗ (xJ ∧ yL).(4.3.22)

where the sum runs over all partitionsI ∪ J = a,K ∪ L = b such that|J | = |K|.

Lemma 4.3.23.Suppose thatΛa+1A M = 0 for some integera ≥ 0. Let 0 < b ≤ a and

x1, ..., xa, y1, ..., yb ∈M∗. Then the following identity holds inΛaAM ⊗A ΛbAM :

xa ⊗ yb =∑

I,J

εJI · (xJ ∧ yb)⊗ xI

where the sum ranges over all the partitionsa = I ∪ J such that|I| = b.

Proof. For a given subsetB ⊂ b we let

γ(yB) :=∑

I,J

εIJ · (xI ∧ yB)⊗ xJ − xa ⊗ yB

where the sum is taken over all the partitionsI ∪ J = a such that|J | = |B|. Notice thatγ(y∅) = 0. We have to show thatγ(yb) = 0. To this purpose we show the following:

Claim4.3.24. If |B| > 0, then

∆a,|B|(xa ∧ yB) =∑

K,L

εKL · (−1)|K| · γ(yK) ∧ yL(4.3.25)

where the sum ranges over all the partitionsK ∪ L = B.


Proof of the claim:Using (4.3.22), the difference between the two sides of (4.3.25) is seen to beequal to

∑K,L εKL · (−1)

|K| · xa ⊗ (yK ∧ yL) =∑

K,L ·(−1)|K| · xa ⊗ yB, where the sum runs

over all partitionsK ∪ L = B. A standard combinatorial argument shows that this expressioncan be rewritten asxa ⊗ yB ·

∑|B|k=0(−1)

k ·(|B|k

), which vanishes if|B| > 0.

To conclude the proof of the lemma, we remark that∆a,b vanishes ifb > 0 because byassumptionΛa+1

A M = 0; then the claim follows by induction on|B|, using claim 4.3.24.

Lemma 4.3.26.LetP be anA-module such thatΛn+1A P = 0 and assume that eitherP is flat

or 2 is invertible inA∗. ThenΛ2A(Λ

nAP ) = 0.

Proof. For anyA-moduleM andr ≥ 0 there exists an antisymmetrizer operator (cp. [10, Ch.III, §7.4, Remarque])

ar :M⊗r →M⊗r m1 ⊗ ...⊗mr 7→

∑

σ∈Sr

sgn(σ) ·mσ(1) ⊗ ...⊗mσ(r).

Clearlyar factors thorughΛrAM , and in caseM is free of finite rank, it is easy to check (justby arguing with basis elements) that the induced mapar : Λ

rAM → Im(ar) is an isomorphism.

This is still true also in caser! is invertible inA∗, since in that case one checks thatar/r! isidempotent (seeloc. cit.). More generally, ifM is flat then, by [38, Ch.I, Th.1.2],M! is thefiltered colimit of a direct system of freeA∗-modules of finite rank, so also in this casear is anisomorphism. Notice that, again by [38, Ch.I, Th.1.2], ifP is flat, thenΛkAP is also flat, for everyk ≥ 0. Hence, to prove the lemma, it suffices to verify thatIm(a2 : (Λ

nAP )

⊗2 → (ΛnAP )⊗2) = 0

whenΛn+1A P = 0. However, this follows easily from lemma 4.3.23.

We are now ready to return toA-modules of finite rank.

Proposition 4.3.27.LetP be an almost projectiveA-module of finite rank; say thatΛrAP = 0.There exists a natural decompositionA ≃ A0 ×A1 × ...×Ar−1 such thatPi := P ⊗A Ai is anAi-module of constant rank equal toi for everyi = 0, ..., r − 1.

Proof. We proceed by induction onr; the caser = 2 is covered by lemma 4.3.12. By lemma4.3.26 we haveΛ2

A(Λr−1A P ) = 0, so by lemma 4.3.12, there is a decompositionA ≃ A′r−1 ×

A′r−2, such that forPi := P ⊗AA′i (i = r− 2, r− 1) the following holds.Λr−1A′r−2

(Pr−2) = 0 and

Λr−1A′r−1

(Pr−1) is an invertibleAr−1-module. It follows in particular thatχPr−1(X) is a polynomial

of degreer−1, and its leading coefficient is invertible inAr−1. HenceχPr−1(X) = (1+X)r−1.By induction,A′r−2 admits a decompositionA′r−2 ≃ Ar−2 × ...×A0 with the stated properties;it suffices then to takeAr−1 := A′r−1.

Theorem 4.3.28.Let P be an almost projectiveA-module of almost finite rank. Then thereexists a natural decompositionA ≃

∏∞i=0Ai such that:

(i) limi→∞

AnnV a(Ai) = V a (for the uniform structure of definition 2.3.2);

(ii) for i ∈ N, let Pi := P ⊗A Ai; thenP ≃∏∞

i=0 Pi and everyPi is anAi-module of finiteconstant rank equal toi.

Proof. Let mλλ∈I be the filtered family of finitely generated subideals ofm. For everyλ ∈ I,letAλ := A/AnnA(mλ). By hypothesis,Pλ := P⊗AAλ is anAλ-module of finite rank; say thatthe rank isr(λ). By proposition 4.3.27 we have natural decompositionsAλ ≃ Aλ,0×...×Aλ,r(λ)such thatP ⊗A Aλ,i is anAλ,i-module of constant rank equal toi for every i ≤ r(λ). Thenaturality of the decomposition means that for everyλ, µ ∈ I such thatmλ ⊂ mµ, we haveAµ,i ⊗A Aλ ≃ Aλ,i. for everyi ≤ r(µ). In particular,mλ · Aµ,i = 0 for everyi > r(λ). Byconsidering the short exact sequence of cofiltered systems of A-modules:

0→ (AnnA(mλ))λ∈I → (A)λ∈I → (Aλ)λ∈I → 0


we deduce easily thatA ≃ limλ∈I

Aλ and therefore we obtain a decompositionA ≃∏∞

i=0Ai,

with Ai := limλ∈I

Aλ,i for every i ∈ N. Notice that, for everyi ∈ N and everyλ ∈ I, the

natural morphismAi → Aλ,i is surjective with kernel killed bymλ. It follows easily thatΛi+1Ai

(P ⊗A Ai) = 0 andΛiAi(P ⊗A Ai) is an invertibleAi-module. Furthermore, for every

λ ∈ I, mλ · Ai = 0 for all i > r(λ), which implies (i). Finally, for everyλ ∈ I, mλ kills thekernel of the projectionP →

∏r(λ)i=0 Pi, soP is isomorphic to the infinite product of thePi.

4.4. Localisation in the flat site. Throughout this sectionP denotes an almost finitely gen-erated projectiveA-module. The following definition introduces the main tool used in thissection.

Definition 4.4.1. Thesplitting algebraof P is theA-algebra:

Split(A, P ) := Sym•A(P ⊕ P∗)/(1− ζP ).

We endowSym•A(P ⊕ P∗) with the structure of graded algebra such thatP is placed in degree

one andP ∗ in degree−1. ThenζP is a homogeneous element of degree zero, and consequentlySplit(A, P ) is also a gradedA-algebra.

4.4.4.4.2. We define a functorS : A-Alg → Set by assigning to everyA-algebraB the setS(B) of all pairs(x, φ) wherex ∈ (P ⊗A B)∗, φ : P ⊗A B → B such thatφ(x) = 1.

Lemma 4.4.3. (i) Split(A, P ) is a flatA-algebra.

(ii) Split(A′, P ⊗A A′) ≃ Split(A, P )⊗A A′ for everyA-algebraA′.(iii) Split(A, P ) represents the functorS.

Proof. For everyk ≥ 0 we haveζkP ∈ (SymkAP )⊗A (Symk

AP∗) ⊂ gr0(Sym2k

A (P ⊕ P ∗)). It iseasy to verify the formula:

grkSplit(A, P ) ≃ colimj∈Z

(Symk+jA P )⊗A (Symj

AP∗)(4.4.4)

where the transition maps in the direct system are given by multiplication by ζP . In particular,it is clear thatgrkSplit(A, P ) is a flatA-module, so (i) holds. (ii) is immediate. To show (iii),let us introduce the functorT : A-Alg → Set that assigns to everyA-algebraB the set of allpairs(x, φ) wherex ∈ (P ⊗A B)∗ andφ : P ⊗A B → B. SoS is a subfunctor ofT .

Claim4.4.5. The functorT is represented by theA-algebraSym•A(P ⊕ P∗).

Proof of the claim:Indeed, there are natural bijections:

(P ⊗A B)∗∼−→ HomB(P

∗ ⊗A B,B)∼−→ HomA(P

∗, B)∼−→ HomA-Alg(Sym

•AP∗, B)

which show that the functorB 7→ (P⊗AB)∗ is represented by theA-algebraSymAP∗. Working

out the definitions, one finds that the composition of these bijections assigns to an elementx ∈ (P ⊗A B)∗ the uniqueA-algebra morphismfx : Sym•AP

∗ → B such thatfx(ψ) = ψ(x)for everyψ ∈ HomA(P,A). Similarly, the functorB 7→ HomB(P ⊗A B,B) is representedby Sym•AP , and again, one checks that the bijection assigns toφ : P ⊗A B → B the uniqueA-algebra morphismgφ : SymAP → B such thatgφ(p) = φ(p) for everyp ∈ P∗. It followsthatT is represented by(Sym•AP )⊗A (Sym•AP

∗) ≃ Sym•A(P ⊕ P∗).

For everyA-algebraB we have a natural map:αB : T (B) → B∗ given by(x, φ) 7→ φ(x).This defines a natural transformation of functorsα : T → (−)∗. Moreover, let us consider thetrivial map βB : T (B) → B∗ that sends everything onto the element1 ∈ B∗. β is anothernatural transformation fromT to the almost elements functor. Clearly :

S(B) = Equal( T (B)αB //

βB

// B∗ ).


We remark that the functorB 7→ B∗ onA-algebras, is represented byA[X ] := Sym•AA. There-fore there are morphismsα∗, β∗ : A[X ] → Sym•A(P ⊕ P

∗) that represent these natural trans-formations. It follows thatS is represented by theA-algebra

Coequal( A[X ]α∗

//

β∗// Sym•A(P ⊕ P

∗) ).

To determineα∗ andβ∗ it suffices to calculate them on the elementX ∈ A[X ]∗. It is easy tosee thatα∗(X) = 1. To conclude the proof it suffices therefore to show:

Claim 4.4.6. β∗(X) = ζP .

Proof of the claim: In view of the definitions, and using the notation of the proofof claim4.4.5, the claim amounts to the identity:(gφ ⊗ fx)(ζP ) = φ(x) for every(x, φ) ∈ T (B). Bynaturality, it suffices to show this forB = A. Now, for everyε ∈ m, we can writeε · ζP =∑

i qi ⊗ ψi for someqi ∈ P , ψi ∈ P ∗ and we have∑

i qi · ψi(b) = ε · b for all b ∈ P∗. Hence(gφ ⊗ fx)(

∑i qi ⊗ ψi) =

∑i φ(qi) · ψi(x) = φ(

∑i qi · ψi(x)) = φ(εx) and the claim follows.

Remark 4.4.7. The construction of the splitting algebra occurs already, in a tannakian context,in Deligne’s paper [15] : see the proof of lemma 7.15 inloc.cit.

4.4.4.4.8. We recall that for everyk ≥ 0 there are natural morphisms

SymkAP

αP−→ ΓkAPβP−→ Symk

AP

such thatβP αP = k! · 1SymkAP

andαP βP = k! · 1ΓkAP

. (to obtain the morphisms, one canconsider the flatA∗-moduleP!, thus one can assume thatP is a module over a usual ring; thenαP is obtained by extending multiplicatively the identity morphismSym1

AP = P → P = Γ1AP ,

andβP is deduced from the homogeneous degreek polynomial lawP ⊗A B → (P ⊗A B)⊗k

defined byx 7→ x⊗k). Moreover(ΓkAP )∗ ≃ Symk

AP∗.

Lemma 4.4.9. With the above notation we have:(αP ⊗A 1SymkAP

∗)(ζkP ) = k! · ζΓkAP

.

Proof. Suppose first thatP is a freeA-module, lete1, ..., en be a base ofP∗ ande∗1, ..., e∗n the

dual base ofP ∗. ThenΓkAP is the freeA-module generated by the basise[n1]1 · ... · e[nk]

k where0 ≤ ni ≤ k for i = 1, ..., k and

∑j nj = k. The dual of this basis is the basis ofSymk

AP∗

consisting of the elementse∗n11 · ... · e∗nk

k . Furthermore,ζP =∑

i ei ⊗ e∗i and thereforeζkP =∑

n

(kn

)(en1

1 · ... · enkk )⊗ (e∗n1

1 · ... · e∗nkk ), wheren := (n1, ..., nk) ranges over the multi-indices

submitted to the above conditions and(kn

):= k!

n1!·...·nk!. Then the claim follows straightforwardly

from the identity:αP (en11 · ... · e

nkk ) = n1! · ... ·nk! · e

[n1]1 · ... · e[nk]

k . For the general case we shalluse the following

Claim 4.4.10. LetM be an almost finitely generated projectiveA-module and pick, for a givenε ∈ m, morphismsu :M → F andv : F →M with v u = ε ·1M . Thenv⊗u∗(ζF ) = ε · ζM .

Proof of the claim:We have a commutative diagram

F ⊗A F ∗v⊗u∗ //

M ⊗AM∗

EndA(F ) // EndA(M)

where the vertical morphisms are the natural ones, and wherethe bottom morphism is given byφ 7→ v φ u. Then the claim follows by an easy diagram chase.


Pick morphismsu : P → F andv : F → P with vu = ε·1P . We consider the commutativediagram

SymkAF ⊗A Symk

AF∗

SymkAv⊗Sym

kAu

∗

//

αF⊗1SymkA

F∗

SymkAP ⊗A Symk

AP∗

αP⊗1SymkA

P∗

ΓkAF ⊗A SymkAF∗

ΓkAv⊗Sym

kAu

∗

// ΓkAP ⊗A SymkAP∗.

By claim 4.4.10, we havev ⊗ u∗(ζF ) = ε · ζP ; whenceSymkAv ⊗ Symk

Au∗(ζkF ) = εk · ζkP .

Moreover, we remark that(ΓkAv) (SymkAu∗)∗ = (ΓkAv) (Γ

kAu) = εk · 1Γk

AP, therefore claim

4.4.10 (applied forM = ΓkAP ) yieldsΓkAv⊗SymkAu∗(ζΓk

AF) = εk ·ζΓk

AP. Since we already know

the lemma forF , a simple diagram chase shows that(α⊗A 1SymkAP

∗)(εk · ζkP ) = εk · k! · ζΓkAP

.Since thek-powers of elements ofm generatem, the claim follows.

Lemma 4.4.11.LetP be as above and suppose thatζP is nilpotent inSym•A(P ⊕ P∗).

(i) If Q ⊂ A∗, thenχP (X) = (1 +X)−α for some continuous functionα : Spec(A∗)→ N.(ii) If Fp ⊂ A∗, thenEP/A is a Frobenius-nilpotent ideal.

Proof. (i): SinceQ ⊂ A∗, thenk! is invertible inA∗ for everyk ≥ 0; by lemma 4.4.9 itfollows thatSymk

AP = 0, thusψP (X) is a polynomial. By corollary 4.3.8,ψP (−X) definesan endomorphism ofGm,A∗ , thereforeψP (X) = (1 + X)α for some continuous functionα :Spec(A∗)→ N; then the claim follows by corollary 4.3.8(ii).

(ii): Let (f, p) ∈ (P ∗ ⊕ P )∗; in the notation of the proof of lemma 4.4.3, we can write(f, p) ∈ T (A). It follows that(f, p) corresponds to a morphism ofA-algebrasSym•A(f, p)

∗ :Sym•A(P ⊕ P

∗)→ A. In particularSym•A(f, p)∗(ζkP ) = Sym•A(f, p)

∗(ζP )k for everyk ≥ 0. By

inspecting the proof of claim 4.4.6, we deduceSym•A(f, p)∗(ζP ) = f(p) for everyf ∈ P ∗ and

p ∈ P . By hypothesis,ζpn

P = 0 for every sufficiently largen. It follows thatEP/A is Frobeniusnilpotent.

Lemma 4.4.12.LetR0 be a noetherian commutative ring,R anR0-algebra andM a flatR-module. ThenM = 0 if and only ifM ⊗R0 κ = 0 for every residue fieldκ ofR0 (i.e., for everyfieldκ of the formFrac(R0/p), wherep is some prime ideal ofR0).

Proof. Clearly we have only to show the direction⇐. It suffices to show thatMp = 0 for everyprime ideal ofR0. Hence we can assume thatR0 is local, in particular of finite Krull dimension.We proceed by induction on the dimension ofR0. If dimR0 = 0, thenR0 is a local artinianring, hence a power of its maximal idealm is equal to0. By assumption,M/m ·M = 0, i.e.M = m ·M . ThenM = mk ·M for everyk ≥ 0, soM = 0. Next, suppose thatdimR0 = dand the lemma already known for all rings of dimension strictly less thand. Assume first thatR0 is an integral domain and pickf ∈ S := R0 \ 0. ThenR0/f · R0 has dimension strictlyless thand, so by induction we haveM/f ·M = 0, i.e.M = f ·M . Due to the flatness ofM ,we have:AnnM(f) = AnnR(f) ·M = AnnR(f) · f ·M = 0. This implies that the kernel ofthe natural mapM → S−1M is trivial. On the other hand, by hypothesisS−1M = 0, whenceM = 0 in this case. For a generalR0 of dimensiond, notice that the above argument impliesthat, for every minimal prime idealp of R0, we havep ·M =M . But the product of all (finitelymany) minimal prime ideals is contained in the nilpotent radicalR of R0, whenceR ·M =M ,and finallyM = 0 as claimed.

4.4.4.4.13. Let nowp ∈ Spec(A∗). By composingf.rkA(P ) with the mapA∗ → A∗/p, weobtain a ring homomorphism

f.rkA(P, p) : G∞ → A∗/p.


In case(A∗/p)a 6= 0, we can interpretf.rkA(P, p) as the formal rank ofP ⊗A (A/pa). Moreprecisely, letπ : A → A/pa be the natural projection; thenπ∗ factors through a mapπ′ :A∗/p→ (A/pa)∗ and we have:f.rkA/pa(P ⊗A (A/pa)) = π′ f.rkA(P, p).

Even if (A∗/p)a = 0, the morphismf.rkA(P, p) can still be interpreted as the map associatedto an endomorphism ofGm,A∗/p, so it still makes sense to ask whetherf.rkA(P, p) is an integer,as indicated in remark 4.3.10(ii).

Lemma 4.4.14.LetP be an almost finitely generated projectiveA-module andp ∈ Spec(A∗).If R is anyAa∗p-algebra andq ∈ Spec(R∗) such thatr(q) := f.rkR(P ⊗A R, q) is an integer,thenr(p) := f.rkA(P, p) is also an integer andr(p) = r(q).

Proof. Indeed, let us consider the natural mapsG∞ → A∗ → R∗; under the assumptions, thecontraction ofq in A∗ is contained inp. Since the image of

(αi

)in R∗/q is

(r(q)i

), it follows that

the same holds inA∗/p.

Definition 4.4.15. We say that anA-moduleP admits infinite splittingsif there is an infinitechains of decompositions of the form:P ≃ A⊕ P1, P1 ≃ A⊕ P2, P2 ≃ A⊕ P3, ...

Theorem 4.4.16.Let P be an almost finitely generated projectiveA-module. The followingconditions are equivalent:

(i) P is of almost finite rank.(ii) For all A-algebrasB 6= 0, we have:

⋂r>0 EΛr

B(P⊗AB)/B = 0.(iii) For all A-algebrasB 6= 0, PB := P ⊗AB does not admit infinite splittings, and moreover

if PB ≃ Bn ⊕ Q for someB-moduleQ andχQ(X) = (1 + X)−α for some continuousfunctionα : Spec(B∗)→ N, thenQ = 0.

(iv) For all A-algebrasB 6= 0, PB does not admit infinite splittings, and moreover ifPB ≃Bn ⊕Q for someB-moduleQ, then :(a) If Fp ⊂ B∗ andQ = I ·Q for a Frobenius-nilpotent idealI ⊂ B, thenQ = 0;(b) If Q ⊂ B∗ andSymr

BQ = 0 for somer ≥ 1, thenQ = 0.

Proof. (i) ⇒ (ii) : indeed, from proposition 2.4.23(iii) one sees that, for everyA-moduleof almost finite rank, everyA-algebraB and everyε ∈ m, there existsr ≥ 0 such thatε · EΛr

B(P⊗AB)/B = 0.(ii) ⇒ (iii) : let B 6= 0 be anA-algebra; by hypothesis, there existsr ≥ 0 such thatJr :=

EΛrB(P⊗AB)/B 6= B. Suppose thatP ⊗A B admits infinite splittings. TheB/Jr-modulePB/Jr ·

PB has rank< r, and at the same time it admits infinite splittings, a contradiction.Suppose next, that there is a decompositionP⊗AB ≃ Bn⊕Q; thenQ is obviously of almost

finite rank. Suppose thatχQ(X) = (1 +X)α has the shape described in (ii). We reduce easilyto the case whereα is a constant function. However,χQ/Jr·Q(X) is a polynomial of degree< r,thusα = 0, and thenQ = 0 by theorem 4.3.28(ii).

(iii) ⇒ (iv) : suppose thatFp ⊂ B∗ andQ = I ·Q for some Frobenius-nilpotent idealI. ThenχQ(X) ∈ 1 + I∗[[X ]], which means that the image ofχQ(X) in End(Gm,B∗/I∗) is the trivialendomorphism. But we have a commutative diagram

C 0(SpecB∗,Zp) //

End(Gm,B∗)

C 0(SpecB∗/I∗,Zp) // End(Gm,B∗/I∗)

where the horizontal maps are those defined in remark 4.3.10(ii), and are bijective by corollary4.2.22. The left vertical map is induced by restriction to the closed subsetSpecB∗/I∗, and


sinceI is Frobenius-nilpotent, it is a bijection as well. It follows that the right vertical map isbijective, whencef.rkB(Q) = 0, and finallyQ = 0 by (iii).

Next, consider the case whenQ ⊂ B∗ andSymrBQ = 0 for somer ≥ 1. It follows thatζQ

is nilpotent inSym•B(Q⊕Q∗). By lemma 4.4.11(i),χQ(X) = (1 +X)−α for some continuous

α : Spec(B∗)→ N. Then (iii) implies thatQ = 0.To show that (iv)⇒ (i), we will use the following:

Claim4.4.17. Assume (iv). Then:Split(B,Q) = 0⇒ Q = 0.

Proof of the claim:SupposeQ 6= 0 andSplit(B,Q) = 0; then (4.4.4) implies that for everyε ∈ m there existsj ≥ 0 such thatε · ζjQ = 0. We haveε · Q 6= 0 for someε ∈ m. Fromthe flatness ofQ, we deriveAnnSym•

B(Q⊕Q∗)(ε) = AnnB(ε) · Sym•B(Q ⊕ Q∗), hence we can

replaceB by B/AnnB(ε), Q by Q/AnnB(ε) · Q, thereby achieving thatζQ is nilpotent inSym•B(Q ⊕ Q∗) and stillQ 6= 0. Using lemma 4.4.12 (and the functoriality ofSplit(B,Q)for base extensionsB → B′) we can further assume thatB∗ contains eitherQ or one of thefinite fieldsFp. If Q ⊂ B∗, thenk! is invertible inB∗ for everyk ≥ 0; by lemma 4.4.9 itfollows thatSymk

BQ = 0, whenceQ = 0 by (iv), a contradiction. IfFp ⊂ B∗, then by lemma4.4.11(ii),EQ/B is Frobenius-nilpotent. However, from proposition 2.4.23(i) it follows easilythatQ = EQ/B ·Q, whenceQ = 0, again by (iv), and again a contradiction. In either case, thisshows thatSplit(B,Q) 6= 0, as claimed.

Claim4.4.18. Assume (iv). ThenA/EP/A is a flatA-algebra.

Proof of the claim:It suffices to show thatAp/(EP/A)p is a flatAp-algebra for every prime idealp ⊂ A∗. If EP/A∗ is not contained inp, then(EP/A)p = Ap, so there is nothing to prove in thiscase. We assume therefore that

EP/A∗ ⊂ p.(4.4.19)

We will show thatPp = 0 in such case, whenceAp/(EP/A)p = Ap, so the claim will follow.From (4.4.19) and corollary 4.3.8(iii) we know already that

f.rkA(P, p) = 0.(4.4.20)

Suppose thatPp 6= 0; then there existsε ∈ m such thatε ·Pp 6= 0. Define inductivelyA0 := Ap,Q0 := Pp, Ai+1 := Split(Ai, Qi) andQi+1 as anAi+1-module such thatQi ⊗Ai

Ai+1 ≃Ai+1 ⊕ Qi+1, for everyi ≥ 0 (the existence ofQi+1 is assured by lemma 4.4.3(iii)). Thencolimn∈N

An ≃ 0, since, after base change to thisV a-algebra,P admits infinite splittings. This

implies that there existsn ∈ N such thatε · An+1 = 0 andε · An 6= 0. However, sinceAn+1

is flat overAn, we have:An+1 = AnnAn+1(ε) = AnnAn(ε) · An+1. SetA′ := An/AnnAn(ε);thenSplit(A′, Qn ⊗An A

′) = 0, soQn ⊗An A′ = 0 by claim 4.4.17. By flatness ofQn, this

means thatε · Qn = 0; in particular,n > 0. By definition,Qn−1 ⊗An−1 An ≃ An ⊕ Qn;it follows thatQ′ := Qn−1 ⊗An−1 A

′ ≃ A′, in particularf.rkA′(Q′) = 1 and consequentlyf.rkA′(P ⊗A A′) = n > 0; in view of lemma 4.4.14, this contradicts (4.4.20), thereforePp = 0,as required.

Claim4.4.21. Assuming (iv), the natural morphismφ : A→ (A/EP/A)× Split(A, P ) is faith-fully flat.

Proof of the claim:The flatness is clear from claim 4.4.18. Hence, to prove the claim, it sufficesto show that, for every proper idealJ ⊂ A, ((A/EP/A)× Split(A, P ))⊗A (A/J) 6= 0. But theconstruction ofφ commutes with arbitrary base changesA → A′, therefore we are reduced toverify that(A/EP/A)× Split(A, P ) 6= 0 whenA 6= 0. By claim 4.4.17, this can fail only whenP = 0; but in this caseEP/A = 0, so the claim follows.


We can now conclude the proof of the theorem: define inductively as in the proof of claim4.4.18:A0 := A,Q0 := P ,Ai+1 := Split(Ai, Qi) andQi+1 as anAi+1-module such thatQi⊗Ai

Ai+1 = Ai+1 ⊕ Qi+1, for everyi ≥ 0. The same argument as inloc. cit. shows that, for everyε ∈ m, there existsn ∈ N such thatε ·An = 0. We may assume thatε ·An−1 6= 0. Moreover, byclaim 4.4.21 (and an easy induction),B := A0/EQ0/A0

×A1/EQ1/A1×...×An−1/EQn−1/An−1

×Anis a faithfully flatA-algebra. However, one checks easily by induction thatP ⊗A (Ai/EQi/Ai

)is a freeAi/EQi/Ai

-module of ranki, for everyi < n. Hence,ΛnB(P ⊗A B) ≃ ΛnAn(P ⊗A An),

which is therefore killed byε. By faithful flatness, so isΛnAP . The proof is concluded.

Proposition 4.4.22. If P is a faithfully flat almost projectiveA-module of almost finite rank,thenSplit(A, P ) is faithfully flat overA.

Proof. If A = 0 there is nothing to prove, so we assume thatA 6= 0. In this case, it suffices toshow thatSplit(A, P ) ⊗A A/I 6= 0 for every proper idealI of A. However,Split(A, P ) ⊗AA/I ≃ Split(A/I, P/I · P ), and sinceP is faithfully flat,P/I · P 6= 0; hence we are reducedto showing thatSplit(A, P ) 6= 0 whenP is faithfully flat. Suppose thatSplit(A, P ) = 0; then(4.4.4) implies that for everyε ∈ m there existsj ≥ 0 such thatε · ζjP = 0. SinceP 6= 0,we haveε · P 6= 0 for someε ∈ m. From the flatness ofP , we deriveAnnSym•

A(P⊕P ∗)(ε) =AnnA(ε) ·Sym

•A(P⊕P

∗), hence we can replaceA byA/AnnA(ε), P byP/AnnA(ε) ·P , whichallows us to assume thatζP is nilpotent inSplit(A, P ). Using lemmata 4.4.12 and 4.4.3(ii), wecan further assume thatA∗ contains eitherQ or one of the finite fieldsFp. If Q ⊂ A∗, thenk!is invertible inA∗ for everyk ≥ 0; by lemma 4.4.9 it follows thatSymk

AP = 0, whenceP = 0by theorem 4.4.16(iv), which contradicts our assumptions,so the proposition is proved in thiscase. Finally, suppose thatFp ⊂ A∗, then by lemma 4.4.11(ii),EP/A is Frobenius-nilpotent.However, sinceP is faithfully flat, proposition 2.4.23(iv) says thatEP/A = A, soA = 0, whichagain contradicts our assumptions.

4.4.4.4.23. For anyV a-algebraA we have a (large) fpqc site on the category(A-Alg)o (insome fixed universe!); as usual, this site is defined by the pretopology whose covering familiesare the finite families(B → Ci)

o | i = 1, ..., n such that the induced morphismB → C1 ×...× Cn is faithfully flat.

Theorem 4.4.24.Every almost projectiveA-module of finite rank is locally free of finite rankin the fpqc topology of(A-Alg)o.

Proof. We iterate the construction ofSplit(A, P ) to split off successive free submodules of rankone. We use the previous characterization of modules of finite rank (proposition 4.3.27) to showthat this procedure stops after finitely many iterations. Byproposition 4.4.22, the output of thisprocedure is a faithfully flatA-algebra.

Theorem 4.4.24 allows to prove easily results on almost projective modules of finite rank, byreduction to the case of free modules. Here are a few examplesof this method.

Lemma 4.4.25.Let P be an almost projectiveA-module of constant rank equal tor ∈ N.Then, for every integer0 ≤ k ≤ r, the natural morphism

ΛkAP ⊗A Λr−kA P → ΛrAP x⊗ y 7→ x ∧ y(4.4.26)

is a perfect pairing.

Proof. By theorem 4.4.24, there exists a faithfully flatA-algebraB such thatPB := P ⊗A Bis a freeB-module of rankr. It suffices to prove the assertion for theB-modulePB, in whichcase the claim is well known.


4.4.4.4.27. Keep the assumptions of lemma 4.4.25. Takingk = 1 in (4.4.26), we derive anatural isomorphism

βP : (Λr−1A P )∗∼→ P ⊗A (ΛrAP )

∗.

Now, let us consider anA-linear morphismφ : P → Q of A-modules of constant rank equal tor. We set

ψ := βP (Λr−1A φ)∗ β−1Q : Q⊗A (ΛrAQ)

∗ → P ⊗A (ΛrAP )∗.

Proposition 4.4.28.With the notation of(4.4.4.4.27), we have:

ψ (φ⊗A 1(ΛrAQ)∗) = 1P ⊗A (ΛrAφ)

∗ and (φ⊗A 1(ΛrAP )∗) ψ = 1Q ⊗A (ΛrAφ)

∗.

Especially,φ is an isomorphism if and only the same holds forΛrAφ.

Proof. After faithfully flat base change, we can assume thatP andQ are free modules of rankr. Then we recognize Cramer’s rule in the above identities.

To conclude this section, we want to apply the previous results to analyze in some detail thestructure of invertible modules : it turns out that the notion of invertibility is rather more subtlethan for usual modules over rings.

Definition 4.4.29. Let M be an invertibleA-module. ClearlyM ⊗A M is invertible as well,consequently the mapA → EndA(M ⊗A M) → A : a 7→ a · 1M⊗AM is an isomorphism(by the proof of lemma 4.1.5(iii)). Especially, for the transposition endomorphismθM |M ofM ⊗A M : x ⊗ y 7→ y ⊗ x, there exists a unique elementuM ∈ A∗ such thatθM |M =uM · 1M⊗AM . Clearlyu2M = 1. We say thatM is strictly invertibleif uM = 1.

Lemma 4.4.30.For an invertibleA-module the following are equivalent:(i) M is strictly invertible;

(ii) Λ2AM = 0;

(iii) M is of almost finite rank;(iv) there exists a faithfully flatA-algebraB such thatM ⊗A B ≃ B.

Proof. (i) ⇒ (ii): indeed, the conditionuM = 1 says that the antisymmetrizer operatora2 :M⊗2 → M⊗2 vanishes (cp. the proof of lemma 4.3.26); sinceM is flat, (ii) follows.

(ii) ⇒ (iii) and (iv)⇒ (i) are obvious. To show that (iii)⇒ (iv) let us setB := Split(A,M);by proposition 4.4.22B is faithfully flat overA, andB ⊗A M ≃ B ⊕ X for someB-moduleX. ClearlyB ⊗A M is an invertibleB-module, therefore, by lemma 4.1.5(ii), the evaluationmorphism gives an isomorphism(B⊕X)⊗A (B⊕X∗) ≃ B⊕X⊕X∗⊕ (X⊗AX∗) ≃ B. Byinspection, the restriction of the latter morphism to the direct summandB equals the identity ofB; henceX = 0 and (iv) follows.

Lemma 4.4.31. If M is invertible, thentrM/A(1M) = uM .

Proof. Pick arbitraryf ∈M∗∗ ,m,n ∈M∗. Then, directly from the definition ofuM we deducethatf(m) · n = uM · f(n) ·m. In other words,ωM/A(n ⊗ f) = uM · evM/A(n ⊗ f) · 1M . Bylinearity we deduce thatφ = uM · trM/A(φ) · 1M for everyφ ∈ EndA(M). By lettingφ := 1M ,and taking traces on both sides, we obtain:trM/A(1M) = uM · trM/A(1M)2. But sinceM isinvertible,trM/A(1M) is invertible inA∗, whenceuM · trM/A(1M) = 1, which is equivalent tothe sought identity.

Proposition 4.4.32.LetM be an invertibleA-module. Then:

(i) M ⊗AM is strictly invertible.(ii) There exists a natural decompositionA ≃ A1×A−1 whereM ⊗AA1 is strictly invertible,

A−1∗ is aQ-algebra andSym2A−1

(M ⊗A A−1) = 0.


Proof. (i): it is clear thatM⊗n is invertible for everyn. Let σ ∈ Sn be any permutation; it iseasy to verify that the morphismσM : M⊗n → M⊗n : x1 ⊗ x2 ⊗ ...⊗ xn 7→ xσ(1) ⊗ xσ(2) ⊗

...⊗xσ(n) equalsusgn(σ)M ·1M⊗n. Especially, the transposition operator on(M ⊗AM)⊗2 acts viathe permutation:x⊗ y ⊗ z ⊗w 7→ z ⊗w ⊗ x⊗ y whose sign is even. ThereforeuM⊗AM = 1,which is (i).

(ii): It follows from (i) that the antisymmetrizer operatora2 on (M ⊗A M)⊗2 vanishes;a fortiori it vanishes on the quotient(Λ2

AM)⊗2, thereforeΛ2A(Λ

2AM) ≃ Im(a2 : Λ2

AM →Λ2AM) = 0. Then lemma 4.3.12 says that there exists a natural decomposition A ≃ A1 × A−1

such that(Λ2AM)⊗AA1 = 0 and(Λ2

AM)⊗AA−1 is invertible. To show thatA−1∗ is aQ-algebra,it is enough to show thatA−1/p · A−1 = 0 for every primep. Up to replacingA by A/p · A,we reduce to verifying that, ifFp ⊂ A∗ andM is invertible, thenM is of almost finite rank.To this aim, it suffices to verify that the equivalent condition (iv) of theorem 4.4.16 is satisfied.If B 6= 0 andMB := M ⊗A B ≃ B ⊕ X, then the argument in the proof of lemma 4.4.30shows thatX = 0 and thereforeMB does not admit infinite splittings. Finally, it remains onlyto verify condition (a) ofloc. cit. So suppose thatMB ≃ Bn ⊕ Q. If n > 0, we have just seenthatQ = 0; if n = 0, andQ/I · Q = 0 for some idealI, then by the faithfulness ofM (lemma4.1.5(iii)) we must haveI = B; if I is Frobenius nilpotent it follows thatB = 0. Finally,setM−1 := M ⊗A A−1; notice that, sinceA−1∗ is aQ-algebra, the endomorphism group ofGm,A−1∗ is isomorphic toA−1∗, and thereforeχM−1(X) = (1 +X)α, whereα is an element ofA−1∗ which can be determined by looking at the coefficient ofχM−1(X) in degree1. One findsα = trM−1/A−1

(1M−1). In view of lemma 4.4.31, we can rewriteα = uM−1; therefore

trΛ2A−1

M−1/A−1(1Λ2

A−1M−1

) =

(uM−1

2

).(4.4.33)

On the other hand, sinceΛ2A−1

M−1 is an invertibleA−1-module of finite rank, we know that theleft-hand side of (4.4.33) equals1; consequentlyuM−1 = −1. This means that, inM⊗2−1 , theidentity x ⊗ y = −y ⊗ x holds for everyx, y ∈ M−1∗; therefore, the kernel of the projectionM⊗2−1 → Sym2

A−1M−1 contains all the elements of the form2 · x ⊗ y; in other words, multipli-

cation by2 is the zero morphism inSym2A−1

M−1; sinceA−1∗ is aQ-algebra, this at last showsthatSym2

A−1M−1 vanishes, and concludes the proof of the proposition.

4.5. Construction of quotients by flat equivalence relations.

4.5.4.5.1. We will need to recall some generalities on groupoids, which we borrow from [17,Exp. V]. If C is any category admitting fibred products and a final object, aC -groupoidis thedatum of two objectsX0,X1 of C , together with ”source” and ”target” morphismss, t : X1 →X0, an ”identity” morphismι : X0 → X1 and a further ”composition” morphismc : X2 → X1,whereX2 is the fibre product in the cartesian diagram:

X2t′ //

s′

X1

s

X1

t // X0.

The datum(X0, X1, s, t, c, ι) is subject to the following condition. For every objectS of C ,the setX0(S) := HomC (S,X0) is a groupoid, with set of morphisms given byX1(S), and foreveryφ ∈ X1(S), the source and target ofφ are respectivelys(φ) := φ s andt(s) := φ s;furthermore the composition law inX0(S) is given byc(S) : X1(S) ×X0(S) X1(S) → X1(S).The above conditions amount to saying thats ι = t ι = 1X0 and the commutative diagrams


in C

X2

c //

t′//

s′

X1

s

X2c //

t′

X1

t

X1

s //

t// X0 X1

t // X0

(4.5.2)

are cartesian both for the square made up from the upper arrows and for the square made upfrom the lower arrows (cp. [17, Exp. V§1]).

One says that the groupoidG := (X0, X1, s, t, c, ι) has trivial automorphisms, if the mor-phism(s, t) : X1 → X0 ×X0 is a (categorical) monomorphism. (This translates in categoricalterms the requirement that for every objectS of C , and everyx ∈ X0(S), the automorphismgroup ofx in X0(S) is trivial).

It is sometimes convenient to denote byX ×(α,β) Z the fibre product of two morphismsα : X → Y andβ : Z → Y .

4.5.4.5.3. Given a groupoidG, and a morphismX0 → X ′0, we obtain a new groupoidG×X0

X ′0 by taking the datum(X ′0, X1 ×X0 X′0, s×X0 1X′

0, t×X0 1X′

0, c×X0 1X′

0, ι×X0 1X′

0).

Moreover, suppose thatC admits finite coproducts and that all such coproducts are disjointuniversal (cp. [1, Exp.II, Def.4.5]). Denote byY ∐ Z the coproduct of two objectsY andZ ofC . LetG′ := (X ′0, X

′1, s′, t′, c′, ι′) be another groupoid; one can define a groupoidG ∐ G′ by

taking the datum(X0 ∐X ′0, X1 ∐X ′1, s∐ s′, t∐ t′, c∐ c′, ι∐ ι′).

4.5.4.5.4. In the following we will be concerned with groupoids in the categoryA-Algo,whereA is anyV a-algebra. In this case, we introduce the following terminology.

Definition 4.5.5. Letφ : X → Y be a morphism inA-Algo,G := (X0, X1, s, t, c, ι) aA-Algo-groupoid.

(i) We say thatφ is aclosed imbedding(resp. isalmost finite, resp. isetale, resp. isflat, resp.is almost projective) if the corresponding morphismφo : Y o → Xo is an epimorphismof Y o-modules (resp. enjoys the same property). We say thatφ is anopen and closedimbeddingif it induces an isomorphismX

∼→ Y1 onto one of the factors of a decomposition

Y = Y1 ∐ Y2.(ii) We say thatG is a closed equivalence relationif the morphism(s, t) : X1 → X0 × X0

is a a closed imbedding. We say thatG is flat (resp. etale, resp. almost finite) if themorphisms : X1 → X0 enjoys the same property. We say thatG is of finite rankif Xo

1

is an almost projectiveXo0 -module of finite rank. Furthermore, we setX0/G := (BG)o,

whereBG ⊂ B := Xo0 is the equalizer of the morphismsso andto.

4.5.4.5.6. LetG := (X0, X1, s, t, c, ι) be a groupoid of finite rank inA-Algo. Let B :=Xo

0 , C := Xo1 ; by assumptionC is an almost projectiveB-module of finite rank, hence, by

proposition 4.3.27, there is a decompositionB ≃∏r

i=0Bi such thatCi := C ⊗B Bi is ofconstant rank equal toi, for i = 0, ..., r. SetX0,i := Bo

i .

Lemma 4.5.7. In the situation of(4.5.4.5.6), there is a natural isomorphism of groupoids:

G ≃ (G×X0 X0,1)∐ ...∐ (G×X0 X0,r).

Proof. For everyi ≤ r, let αi : X0,i → X0 be the open and closed imbedding defined by(4.5.4.5.6). SetX1,i := X0,i×(αi,s)X1 (soXo

1,i = Ci). Moreover, letX ′1,i := X0,i×(αi,t)X1, βi :X ′1,i → X1 the open and closed imbedding (obtained by pulling backαi),X2,i := X1,i×(βi,s′)X2

andX2,i := X ′1,i ×(βi,s′) X2. There follow natural decompositionsX1 ≃ X ′1,1 ∐ ... ∐X′1,r and

X2 ≃ X ′2,1 ∐ ... ∐ X′2,r, such thats′ decomposes as a coproduct of morphismsX ′2,i → X ′1,i.

By the construction ofX0,i, it is clear thatX ′o2,i has rank equal toi as anX ′o1,i-module, for every


i ≤ r. In other words, the above decompositions fulfill the conditions of proposition 4.3.27.Similarly, we obtain decompositionsX1 ≃ X1,1 ∐ ... ∐ X1,r andX2 ≃ X2,1 ∐ ... ∐ X2,r

which fulfill the same conditions. However, these conditions characterize uniquely the factorsoccuring in it, thusX1,i = X ′1,i for i ≤ r. The claim follows easily.

Lemma 4.5.8. LetG := (X0, X1, s, t, c, ι) be a groupoid of finite rank inA-Algo. If G hastrivial automorphisms, then it is a closed equivalence relation.

Proof. LetB := Xo0 andC := Xo

1 . Using lemma 4.5.7 we reduce easily to the case whereC isof constant rank, say equal tor ∈ N. LetY := X0×X0; sinceG has trivial automorphisms, themorphism(s, t) : X1 → Y is a monomorphism; equivalently, the natural projectionspr1, pr2 :X1 ×Y X1 → X1 are isomorphisms. LetD := Im(B ⊗A B → C); it follows that the naturalmorphismspro1, pr

o2 : C → C ⊗D C are isomorphisms and consequently,

(C/D)⊗D C = 0.(4.5.9)

We need to show thatC = D, or equivalently, thatC/D = 0. However, by theorem 4.4.24, wecan find a faithfully flatB-algebraB′ such thatC ′ := B′ ⊗B C ≃ (B′)r. LetD′ := B′ ⊗B D;it follows thatC ′ ≃ (B′)r, in particularC ′ is a faithful finitely generatedD′-module. It sufficesto show thatC ′/D′ = 0, and we know already from (4.5.9) that(C ′/D′) ⊗D′ C ′ = 0. Byproposition 3.3.4 it follows thatC ′/D′ is a flatD′-module; consequentlyC ′/D′ ⊂ (C ′/D′)⊗D′

C ′, and the claim follows.

4.5.4.5.10. LetB be anA-algebra,P an almost finitely generated projectiveB-module. Forevery integeri ≥ 0, we define aB-linear morphism

ΓiB(EndB(P )a)→ B(4.5.11)

as follows (see (7.1.7.1.15) for the definition of the functor ΓiB : B-Mod → B-Mod). LetRbe anyB∗-algebra; we remark that the natural mapβP : EndB(P )

a! ⊗B∗R→ EndRa(Ra⊗AP )

a!

is an isomorphism. Hence, for everyi ≥ 0, we can define a map of sets

λiR : EndB(P )a! ⊗B∗ R→ EndB(Λ

iBP )

a! ⊗B∗ R

by lettingφ 7→ β−1ΛiBP

(ΛiRaβP (φ)a). In the terminology of [46], the system of mapsλiR forms

a homogeneous polynomial law of degreei from EndB(P )a! to EndB(Λ

iBP )

a! , so it induces a

B∗-linear mapλi : ΓiB∗(EndB(P )

a! ) → EndB(Λ

iBP )

a! . After passing to almost modules, we

obtain aB-linear morphism

ΓiB(EndB(P )a)→ EndB(Λ

iBP )

a.(4.5.12)

Then (4.5.11) is defined as the composition of (4.5.12) and the trace morphismtrΛiBP/B

.

4.5.4.5.13. LetC be an almost finite projectiveB-algebra. Defineµ : C → EndB(C)a as in

(4.1.4.1.7). By composition ofΓiBµ and (4.5.11) we obtain aB-linear morphism

ΓiBC → B

characterized by the condition:c[i] 7→ σi(c) := trC/B(ΛiCµ(c)). Now one verifies, using the

cartesian diagrams (4.5.2), thatσi(to(f)) ∈ BG for everyf ∈ B∗ and everyi ≤ r: the argument

is the same as in the proof of [17, Exp.V, Th.4.1]. In this way one obtainsBG-linear morphisms

TG,i : ΓiBGB

Γito−→ ΓiBC → BG f [i] 7→ σi(t

o(f)).(4.5.14)

Theorem 4.5.15.LetG := (X0, X1, s, t, c, ι) be anetale almost finite and closed equivalencerelation inA-Algo. ThenG is effective and the natural morphismX0 → X0/G is etale andalmost finite projective.


Proof. See [17, Exp.IV,§3.3] for the definition of effective equivalence relation. By (4.5.2),we have an identificationX2 ≃ X1 ×(s,s) X1.; therefore, the natural diagonal morphismX1 →X1 ×(s,s) X1 gives a sectionδ : X1 → X2 of the morphisms′ : X2 → X1. Furthermore, sinceX2 = X1 ×(s,t) X1, the pair of morphisms(1X1 , ι s) : X1 → X1 induces another morphismψ0 : X1 → X2; similarly, letψ1 : X1 → X2 be the morphism induced by the pair(ι t, 1X1)(these are the degeneracy maps of the simplicial complex associated toG: cp. [17, Exp. V,§1]).By arguing withT -points (and exploiting the interpretation (4.5.4.5.1) ofX0(T ), X1(T ), etc.)one checks easily, first thatψ1 = δ, and second, that the two commutative diagrams

X0ι //

ι

X1

ψ1

X1t //

ψ1

X0

ι

X1

ψ0 // X2 X2t′ // X1

(4.5.16)

are cartesian. Since by assumptions is etale, corollary 3.1.9 implies thatδ is an open andclosed imbedding; consequently the same holds forι. Let e0 ∈ (Xo

1)∗ (resp. e1 ∈ (Xo2)∗)

be the idempotent corresponding to the open and closed imbedding ι (resp. δ); sinceG is aclosed equivalence relation, for everyε ∈ m we can writeε · e0 =

∑ni s

o(bi) · to(b′i) for some

bi, b′i ∈ (Xo

1)∗. In view of (4.5.16) we deduce thatε · e1 =∑n

i (t′o so(bi)) · (t′o to(b′i)).

However,s t′ = t s′ andt t′ = t c, consequently

ε · e1 =n∑

i

(s′o to(bi)) · (co to(b′i)).

Finally, thanks to remark 4.1.17, and again (4.5.2), we can write:

ε · f =

n∑

i

so TrXo1/X

o0(f · to(bi)) · t

o(b′i) for everyf ∈ Xo1 .(4.5.17)

If we now let f := to(g) in (4.5.17) we deduce:ε · to(g) =∑

i so(TG,1(g · bi)) · to(b′i) =∑

i to(TG,1(g · bi)) · to(b′i) for everyg ∈ B. Sinceto is injective, this means that:

ε · g =n∑

i

TG,1(g · bi) · b′i for everyg ∈ B := Xo

0 .(4.5.18)

It follows easily thatB is an almost finitely generated projectiveBG-module. Furthermore, letus introduce the bilinear pairingtG := TG µB/BG : B ⊗BG B → BG.

Claim4.5.19. tG is a perfect pairing.

Proof of the claim: We have to show that the associatedB-linear morphismτG : B →B∗ := alHomBG(B,BG) is an isomorphism. From (4.5.18) it follows easily thatτG is amonomorphism. Letφ : B → BG be aBG-linear morphism; it remains only to show that,for every ε ∈ m, there existsb ∈ B∗ such thatτG(b∗) = ε · φ. SetC := Xo

1 and letα : C → Bn, β : Bn → C be defined by the rules:c 7→ (TrC/B(c · b1), ...,TrC/B(c · bn)) and(x1, ..., xn) 7→

∑ni xi · b

′i for all c ∈ C∗, x1, ..., xn ∈ B∗. We remark thatIm(α to) ⊂ (BG)n

andIm(β (to)n) ⊂ B, so that we deduce, by restriction, morphismsα0 : B → (BG)n andβ0 : (B

G)n → B. Letψ := ((φ β0)⊗BG 1B) α : C → B. By theorem 4.1.14 we can find,for everyε ∈ m, an elementc ∈ C∗ such thatε · ψ = τC/B(c). Using (4.5.17) we derive easilyε · c = ε ·

∑ni ψ(t

o(bi)) · to(b′i) = ε ·∑n

i φ β0 α0(bi) · to(b′i) = ε ·∑n

i φ(ε · bi) · to(bi). In

particular,ε · c = to(b) for someb ∈ B, so the claim follows.

By assumption, the morphismπ : C ′ := B ⊗BG B → C induced by the pair(so, to) is anepimorphism. Moreover, by construction, we have the identity: TrC/B π = 1B ⊗BG TG,1. Byclaim 4.5.19 we see that1B ⊗BG TG,1 induces a perfect pairingC ′ ⊗B C ′ → B; on the other


hand,TrC/B is already a perfect pairing, by theorem 4.1.14. It then follows thatπ must be amonomorphism, henceC ′ ≃ C, which shows thatG is effective; then it is easy to verify thatTG,1 is actually the trace of theBG-algebraB, which is consequently etale overBG.

Proposition 4.5.20.Suppose thatG is of finite rank. ThenB∗ is integral overBG∗ .

Proof. By lemma 4.5.7 we can reduce to the case where the rank ofC := Xo1 is constant, say

equal tor. The assertion is then a direct consequence of the following:

Claim 4.5.21. Let f ∈ B∗. With the notation of (4.5.4.5.13) we have:

(to(f))r + TG,1(f) · (to(f))r−1 + TG,2(f) · (t

o(f))r−2 + ...+ TG,r(f) = 0.

Proof of the claim:By theorem 4.4.24,C is locally free of rankr in the fpqc topology ofB;since everything in sight commutes with base change, we can therefore assume thatC is a freeB-module of rankr. In that case, the sought identity follows from Cayley-Hamilton’s theorem,applied to the endomorphismµ(to(f)) : C∗ → C∗.

Proposition 4.5.22.Let G be anetale closed equivalence relation of finite rank. ThenG isuniversally effective and the morphismX0 → X0/G is etale, faithfully flat and almost finiteprojective.

Proof. Everything is known by theorem 4.5.15, except for the faithfulness, which follows fromthe following:

Claim 4.5.23. Under the assumptions of proposition 4.5.20, and letI ⊂ BG be an ideal suchthatI · B = B. ThenI = BG.

Proof of the claim:First of all, letB ≃∏r

i=0Bi be the decomposition associated, by proposition4.3.27, to theB-moduleXo

1 ; one derives easily a corresponding decompositionBG ≃∏r

i=0BGi ,

so we can assume that the rank ofXo1 is constant, equal tor. Let J ⊂ BG be any ideal, and set

C := BG/J . We have a natural isomorphismΓiC(B ⊗BG C) ≃ ΓiBG(B) ⊗BG C; composingwith (4.5.14)⊗BGC, we derive aC-linear morphism:ψi : ΓiC(B ⊗BG C) → C. By inspectingthe construction, one shows easily thatψi(1

[i]) =(ri

)(indeed, by flat base change one reduces

easily to the case whereXo1 is a freeB-module of rankr, in which case the result is obvious).

Let us now takeJ = I. ThenΓiC(B ⊗BG C) = 0 for everyi > 0, whence1 = ψr(1[r]) = 0 in

C, and the claim follows.

Proposition 4.5.24.Keep the assumptions of proposition 4.5.22. IfXo0 is an almost finite (resp.

almost finitely presented, resp. flat, resp. almost projective, resp. weakly unramified, resp.unramified, resp. weaklyetale, resp.etale)A-algebra, then the same holds for theA-algebra(X0/G)

o.

Proof. SetC := (X0/G)o. By proposition 4.5.22,B := Xo

0 is a faithful almost finitely gener-ated projectiveC-module, henceEB/C = C by proposition 2.4.23(iv). It follows easily that, foreveryε ∈ m there existsn ∈ N such thatε ·1C factors as a composition ofC-linear morphisms:

C → Bn → C.(4.5.25)

The assertions for “almost finite” and for ”almost projective are immediate consequences. Toprove the assertion for “almost finitely presented” we use the criterion of proposition 2.3.14(ii).Indeed, let(Nλ, φλµ | λ) be a filtered system ofA-modules; we apply the natural transformation(2.3.15) to the sequence of morphisms (4.5.25) : sinceB is almost finitely presented, so isBn, hence the claim follows by a little diagram chase. The assertions for “flat” and “weaklyunramified” are easy and shall be left to the reader. To conclude, it suffices to consider the


assertion for “unramified”. Now, by proposition 4.5.22 it follows thatB ⊗A B is an almostfinitely generated projectiveC ⊗A C-module; since by assumptionB is an almost projectiveB ⊗A B-module, we deduce from lemma 2.4.5 thatB is an almost projectiveC ⊗A C-module.Using (4.5.25) we deduce thatC is almost projective overC ⊗A C as well.


5. VALUATION THEORY

5.1. Ordered groups and valuations. In this section we gather some generalities on valua-tions and related ordered groups, which will be used in latersections.

5.1.5.1.1. As usual, avalued field(K, | · |K) consists of a fieldK endowed with a surjectivegroup homomorphism| · |K : K× → ΓK onto an ordered abelian group(ΓK ,≤), such that

|x+ y|K ≤ max(|x|K , |y|K)(5.1.2)

wheneverx+y 6= 0. We denote by1 the neutral element ofΓK , and the composition law ofΓKwill be denoted by:(x, y) 7→ x · y. It is customary to extend the map| · |K to the whole ofK,by adding a new element0 to the setΓK , and setting|0| := 0. One can then extend the orderingof ΓK to ΓK ∪ 0 by declaring that0 is the smallest element of the resulting ordered set. Inthis way, (5.1.2) holds for everyx, y ∈ K. The map| · |K is called thevaluationof K andΓKis itsvalue group.

5.1.5.1.3. Anextension of valued fields(K, | · |K) ⊂ (E, | · |E) consists of a field extensionK ⊂ E, and a valuation| · |E : E → ΓE ∪ 0 together with an imbeddingj : ΓK ⊂ ΓE , suchthat the restriction toK of | · |E equalsj | · |K .

Example 5.1.4.Let | · | : K → Γ ∪ 0 be a valuation on the fieldK.(i) Given a field extensionK ⊂ E, it is known that there always exist valuations onE which

extend| · | (cp. [11, Ch.VI,§1, n.3, Cor.3]).(ii) If the field extensionK ⊂ E is algebraic and purely inseparable, then the extension of

| · | is unique. (cp. [11, Ch.VI,§8, n.7, Cor.2]).(iii) We can construct extensions of| · | on the polynomial ringK[X ], in the following way.

Let Γ′ be an ordered group with an imbedding of ordered groupsΓ ⊂ Γ′. For everyx0 ∈ K,and everyρ ∈ Γ′, we define theGauss valuation| · |(x0,ρ) : K[X ] → Γ′ ∪ 0 centered atx0and with radiusρ (cp. [11, Ch.VI,§10, n.1, Lemma 1]) by the rule:

a0 + a1(X − x0) + ... + an(X − x0)n 7→ max|ai| · ρ

i | i = 0, 1, ..., n.

(iv) The construction of (iii) can be iterated : for instance, suppose that we are given asequence ofk elementsρ := (ρ1, ρ2, ..., ρk) of the ordered abelian groupΓ′ of (iv). Thenwe can define a Gauss valuation| · |(0,ρ) on the fraction field ofK[X1, X2, ..., Xk], with valuesin Γ′, by the rule:

∑α∈Nk aαX

α 7→ max|aα| · ρα | α ∈ N.(v) Suppose again it is given an ordered groupΓ′ with an imbedding of ordered groupsΓ ⊂

Γ′. LetT ⊂ Γ′/Γ be a finite torsion subgroup, sayT ≃ Z/n1Z⊕ ...⊕Z/nkZ. For everyi ≤ k,pick an elementγi ∈ Γ′ whose class inΓ′/Γ generates the direct summandsZ/niZ of T . Letxi ∈ K such that|xi| = ni · γi. For everyi = 1, ..., k pick an elementyi in a fixed algebraicclosureEa of E, such thatyni

i = xi; then the fieldE := K(y1, ..., yk) has degree overK equalto the order ofT , and it admits a unique valuation| · |E extending| · |. Of course,|yi|E = γi foreveryi ≤ k.

5.1.5.1.5. We want to explain a construction which is a simultaneous generalization of theexamples 5.1.4(iv),(v). Suppose it is given the datumG := (G, j,N,≤) consisting of:

(a) an abelian groupG with an imbeddingj : K× → G such thatG/j(V ×) is torsion-free;(b) a subgroupN of G/j(V ×) such that the natural map:

Γ∼→ K×/V × → ΓG := G/(N + j(V ×))

is injective;(c) an ordering≤ onΓG such that the injective map:Γ→ ΓG is order-preserving.


Let us denote byK[G] (resp.K[K×]) the groupK-algebra of the abelian groupG (resp.K×).Any element ofK[G] can be written uniquely as a formal linear combination

∑g∈G ag · [g],

whereag ∈ K for everyg ∈ G, andag = 0 for all but a finite number ofg ∈ G. We augmentK[K×] overK via theK-algebra homomorphism

K[K×]→ K : [a] 7→ a for everya ∈ K×.(5.1.6)

Then we letK[G] := K[G] ⊗K[K×] K, where theK[K×]-algebra structure onK is definedby the augmentation (5.1.6). It is easy to verify thatK[G] is the maximal quotient algebra ofK[G] that identifies the classes of[g · a] anda · [g], for everyg ∈ G anda ∈ K×. Pick, forevery classγ ∈ G/K×, a representativegγ ∈ G. It follows that every element ofK[G] canbe written uniquely as a formalK-linear combination

∑γ∈G/K× aγ · [gγ]. We define a map

| · |G : K[G]→ ΓG ∪ 0 by the rule:∑

γ∈G/K×

aγ · [gγ ] 7→ maxγ∈G/K×

|aγ| · |gγ|(5.1.7)

where|gγ| ∈ ΓG denotes the class ofgγ. One verifies easily that| · |G does not depend on thechoice of representativesgγ. Indeed, if(hγ | γ ∈ G/K×) is another choice then, for everyγ ∈ G/K× we havegγ = j(xγ) · hγ for somexγ ∈ K×; therefore[gγ] = xγ · [hγ] and|gγ| = |xγ | · |hγ|.

Lemma 5.1.8.K[G] is an integral domain, and| · |G extends to a valuation:

| · |G : K(G) := Frac(K[G])→ ΓG∪ 0.

Proof. Let (Gα | α ∈ I) be the filtered system of the subgroupsGα of G such thatK× ⊂ Gα

andGα/K× is finitely generated. EachGα defines a datumGα := (Gα, j, N∩(Gα/j(V

×)),≤),and clearlyK[G] = colim

α∈IK[Gα]. We can therefore reduce to the case whereG/K× is finitely

generated. WriteG/K× = T ⊕F , whereT is a torsion group andF is torsion-free. There existunique subgroupsT , F ⊃ K× inGwith T /K× = T andF /K× = F . LetGT := (T , j, 0,≤)

andGF := (F , j, N,≤) be the corresponding data. The functorH 7→ K[H ] preserves colimits,since it is left adjoint to the forgetful functor fromK-algebras to abelian groups; it followseasily thatK[G] ≃ K[GT ]⊗K K[GF ]. By inspecting (5.1.7), one can easily show thatK[GT ]is of the type of example 5.1.4(v) andK[GF ] is of the type of example 5.1.4(iv). Especially,K[G] is a domain, and| · |G is induced by a Gauss valuation of a free algebra over the finite fieldextensionK[GT ] of K.

The next result shows that an arbitrary valuation is always “close” to some Gauss valuation.

Lemma 5.1.9. Let (E, | · |E) be a valued field extension of(K, | · |). Letx ∈ E \ K, and let(ai | i ∈ I) be a net of elements ofK (indexed by the directed set(I,≤)) with the followingproperty. For everyb ∈ K there existsi0 ∈ I such that|x − ai|E ≤ |x − b|E for everyi ≥ i0.Letf(X) ∈ K[X ] be a polynomial that splits inK[X ] as a product of linear polynomials. Thenthere existsi0 ∈ I such that|f(x)|E = |f(X)|(ai,|x−ai|E) for everyi ≥ i0.

Proof. To prove the claim, it suffices to consider the case whenf(X) = X− b for someb ∈ K.However, from the definition of the sequence(ai | i ∈ I) we havemax(|x − ai|E, |ai − b|) ≥|x−b|E ≥ |x−ai|E for every sufficiently largei ∈ I. Therefore,|x−b|E = max(|x−ai|E, |b−ai|) = |X − b|(ai,|x−ai|E).

5.1.5.1.10. To see how to apply lemma 5.1.9, let us consider the case whereK is algebraicallyclosed andE = K(X), the field of fractions of the freeK-algebra in one generator, whichwe suppose endowed with some valuation| · |E with values inΓE . We apply lemma 5.1.9 tothe elementx := X ∈ E. Suppose first that there exists an elementa ∈ K that minimizes


the functionK → ΓE : b 7→ |X − b|E. In this case the trivial neta fulfills the conditionof the lemma. Since every polynomial ofK[X ] splits overK, we see that| · |E is the Gaussvaluation centered ata and with radius|X − a|E. Suppose, on the other hand, that the functionb 7→ |X − b|E does not admit a minimum. It will still be possible to choose anet of elementsai | i ∈ I fulfilling the conditions of lemma 5.1.9 (indexed, for instance, by a subset of thepartially ordered setΓ). Then| · |E is determined by the identity:

|f(X)|E = limi∈I|f(X)|(ai,|X−ai|E) for everyf(X) ∈ E.

5.1.5.1.11. Given a valuation|·| on a fieldK, the subsetK+ := x ∈ | |x| ≤ 1 is avaluationring of K, i.e., a subring ofK such that, for everyx ∈ K \ 0, eitherx ∈ K+ or x−1 ∈ K+.The subset(K+)× of units ofK+ consists precisely of the elementsx ∈ K such that|x| = 1.Conversely, letV be a valuation ring ofK with maximal idealm; V induces a valuation| · | onK whose value group isΓK := K×/V × (then| · | is just the natural projection). The orderingonΓK is defined as follows. For given classesx, y ∈ ΓK , we declare thatx < y if and only ifx/y ∈ m.

Remark 5.1.12. (i) It follows easily from (5.1.5.1.11) that every finitely generated ideal of avaluation ring is principal. Indeed, ifa1, ..., an is a set of generators for an idealI, pick i0 ≤ nsuch that|ai0 | = maxi≤n |ai|; thenI = (ai0).

(ii) It is also easy to show that any finitely generated torsion-freeK+-module is free and anytorsion-freeK+-module is flat (cp. [11, Ch.VI,§3, n.6, Lemma 1]).

(iii) Let E be a field extension of the valued field(K, | · |K). Then the integral closureW ofK+ in E is the intersection of all the valuation rings ofE containingK+ (cp. [11, Ch.VI,§1,n.3, Cor.3]). In particular,K+ is integrally closed.

(iv) Furthermore, ifE is an algebraic extension ofK, thenW is aPrufer domain, that is, forevery prime idealp ⊂ W , the localizationWp is a valuation ring. Moreover, the assignmentm 7→Wm establishes a bijection between the set of maximal ideals ofW and the set of valuationringsV of E whose associated valuation| · |V extends| · |K (cp. [11, Ch.VI,§8, n.6, Prop.6]).

(v) Let R andS be local rings contained in a fieldK, mR andmS their respective maximalideals. One says thatR dominatesS if S ⊂ R andmS = S ∩mR. It is clear that the relation ofdominance establishes a partial order structure on the set of local subrings ofK. Then a localsubring ofK is a valuation ring ofK if and only if it is maximal for the dominance relation (cp.[11, Ch.VI,§1, n.2, Th.1]).

(vi) Let K+ be a valuation ring ofK with maximal idealm, andK+h a henselization ofK+. One knows thatK+h is an ind-etale localK+-algebra (cp. [45, Ch.VIII, Th.1]), henceit is integral and integrally closed (cp. [45, Ch.VII,§2, Prop.2]). Denote byKh the field offractions ofK+h andW the integral closure ofK+ in Kh. It follows thatW ⊂ K+h. Let mh

be the maximal ideal ofK+h; sincemh ∩K+ = m, we deduce thatq := mh ∩W is a maximalideal ofW ; then by (iv),Wq is a valuation ring ofKh dominated byK+h; by (v) it followsthatK+h = Wq, in particular this shows that the henselization of a valuation ring is again avaluation ring. The same argument works also for strict henselizations.

The following lemma provides a simple method to construct extensions of valuation rings,which is sometimes useful.

Lemma 5.1.13.Let(K, | · |) be a valued field,κ the residue field ofK+,R aK+-algebra whichis finitely generated free as aK+-module, and suppose thatR ⊗K+ κ is a field. ThenR is avaluation ring, and the morphismK+ → R induces an isomorphism of value groupsΓK

∼→ ΓR.

Proof. Let e1, ..., en be aK+-basis ofR. Let us define a map| · |R : R → ΓR ∪ 0 in thefollowing way. Givenx ∈ R, write x =

∑ni=1 xi · ei; then|x|R := max|xi| | i = 1, ..., n.

If |x| = 1, then the imagex of x in R ⊗K+ κ is not zero, hence it is invertible by hypothesis.


By Nakayama’s lemma it follows easily thatx itself is invertible inR. Hence, every elementyof R can be written in the formy = u · b, whereu ∈ R× andb ∈ K+ is an element such that|b| = |y|R. It follows easily thatR is an integral domain. Moreover, it is also clear that, givenanyx ∈ Frac(R) \ 0, eitherx ∈ R or x−1 ∈ R, soR is indeed a valuation ring and| · |R is itsvaluation.

Lemma 5.1.14.Every finitely presented torsionK+-moduleM is isomorphic to a direct sumof the form

(K+/a1 ·K+)⊕ ...⊕ (K+/an ·K

+)

wherea1, ..., an ∈ K+. More precisely, ifFφ→ M is any surjection from a freeK+-module

F of rankn, then there is a basise1, ..., en of F and elementsa1, ..., an ∈ K+ \ 0 such thatKer(φ) = (a1K

+)⊕ ...⊕ (anK+).

Proof. We proceed by induction on the rankn of F . Forn = 1 the claim follows easily fromremark 5.1.12(i). Supposen > 1; first of all,S := Ker(φ) is finitely generated by [11, Ch.I,§2,n.8, lemme 9]. ThenS is a freeK+-module, in light of remark 5.1.12(ii); its rank is necessarilyequal ton, sinceS ⊗K+ K = F ⊗K+ K.

The image of the evaluation mapS ⊗K+ F ∗ → K+ given byf ⊗ α 7→ α(f) is a finitelygenerated idealI 6= 0 of K+, hence it is principal, by remark 5.1.12(i). Let

∑i=1 fi ⊗ αi be an

element whose image generatesI; this means that∑

i=1 αi(fi) is a generator ofI, hence one ofthe terms in the sum, sayα1(f1), is already a generator. The mapα1 : S → I is surjective ontoa free rank oneK+-module, therefore it splits, which shows thatS = (f1K

+)⊕ (S ∩Ker α1).In particular,S ′ := S ∩Ker α1 is a finitely generated torsion-free, hence freeK+-module. Lete1, ..., en be a basis ofF ; thenf1 =

∑ni=1 ai · ei for someai ∈ K+. Consider the projection

πi : F → K+ such thatπi(ej) = δij for j = 1, ..., n; clearlyπi(f1) = ai ∈ I. This shows thatf1 = α1(f1) · g for someg ∈ F . It follows thatα1(g) = 1, whenceF = (gK+)⊕Ker(α1). SetF ′ := Ker(α1); we have shown thatM ≃ (K+/α1(f1))⊕(F ′/S ′). ButF ′ is a freeK+-moduleof rankn− 1, hence we conclude by induction.

5.1.5.1.15. In later sections we will be concerned with almost ring theory in the special casewhere the basic setup(V,m) (see 2.1.2.1.1) consists of a valuation ringV . In preparation forthis, we fix the following terminology, which will stand throughout the rest of this work. IfV isa valuation ring, then thestandard setupattached toV is the pair(V,m) wherem := V in casethe value group ofV is isomorphic toZ (i.e. V is adiscrete valuation ring), and otherwisem isthe maximal ideal ofV .

5.1.5.1.16. LetK → ΓK ∪ 0 be a valuation on the fieldK, andK+ its valuation ring.We consider the categoryK+a-Mod relative to the standard setup(K+,m). The topologicalgroupDiv(K+a) of fractional idealsof K+a is the subspace ofIK+a(Ka) which consists ofall the submodulesI 6= Ka of the almostK+-moduleKa, such that the natural morphismI ⊗K+a Ka → Ka is an isomorphism. The group structure is induced by the multiplication offractional ideals.

Remark 5.1.17. One verifies easily thatDiv(K+a) is isomorphic to the groupD(K+) definedin [11, Ch.VII, §1, n.1].

The structure of the ideals ofK+ can be largely read off from the value groupΓ. In order toexplain this, we are led to introduce some notions for general ordered abelian groups.


5.1.5.1.18. We endow an ordered groupΓ with the uniform structure defined in the followingway. For everyγ ∈ Γ such thatγ > 1, the subset ofΓ × Γ given byE(γ) := (α, β) | γ−1 <α−1 · β < γ is an entourage for the uniform structure, and the subsets ofthis kind form afundamental system of entourages. LetΓ∧ be the completion ofΓ for this uniform structure.

Lemma 5.1.19.With the notation of(5.1.5.1.16), there exists a natural isomorphism of topo-logical groups:Div(K+a)

∼→ Γ∧K .

Proof. We only indicate how to construct the morphism, and leave thedetails to the reader. Inlight of remark 5.1.12(i), for every idealI ⊂ K+a we can find a netJi | i ∈ S of principalideals converging toI (for some filtered ordered set(S,≤)). Let γi ∈ ΓK be the value of agenerator ofJi. One verifies that the netγi | i ∈ S converges inΓ∧K to some elementγ. Thenwe assign:I 7→ γ. One verifies that this rule is well-defined and that it extends uniquely to thewhole ofDiv(K+a).

Definition 5.1.20. Let Γ be any ordered abelian group with neutral element1.(i) We denote byΓ+ ⊂ Γ the subset of all theγ ∈ Γ such thatγ ≤ 1.(ii) A subgroup∆ of Γ is said to beconvexif it satisfies the following property. Ifx ∈ ∆+

and1 > y > x, theny ∈ ∆. The setSpec(Γ) of all the convex subgroups ofΓ will becalled thespectrumof Γ. We define theconvex rankof Γ as the supremumc.rk(Γ) overthe lengthsr of the chains0 ( ∆1 ( ... ( ∆r := Γ, such that all the∆i are convexsubgroups. In generalc.rk(Γ) ∈ N ∪ ∞, but we will mainly encounter situations forwhich the convex rank is a positive integer. It is easy to see that the convex rank is alwaysless than or equal to the usual rank, defined asrk(Γ) := dimQ(Γ⊗Z Q). To keep the twoapart, we callrational rank the latter.

Example 5.1.21.(i) If Γ is an ordered abelian group, there exists a unique ordered group struc-ture onΓ ⊗Z Q such that the natural mapΓ → Γ ⊗Z Q is order-preserving. Indeed, ifΓ is thevalue group of a valuation| · | on a fieldK, and| · |Ka is any extension of| · |K to the algebraicclosureKa of K, then it is easy to see (e.g.using example 5.1.4(v)) thatΓKa ≃ Γ⊗Z Q.

(ii) Furthermore, letKs ⊂ Ka be the separable closure ofK; we claim that| · |Ka mapsKs

surjectively ontoΓKa. Indeed, ifa ∈ Ka is inseparable overKs, then the minimal polynomialm(X) ∈ Ks[X ] of a is of the formXpm − b for someb ∈ Ks. For c ∈ K×, let mc(X) ∈Ks[X ] be the polynomialm(X) + c · X; if a′ is a root ofmc(X), thena′ ∈ Ks; moreover,|(a− a′)p

m|Ka = |c · a′|Ka, hence for|c|Ka sufficiently small we have|a|Ka = |a′|Ka.

(iii) For any valued field(K, | · |), and everyγ ∈ ΓK , let Uγ := x ∈ K | |x| ≤ γ.One defines thevaluation topologyon K as the unique group topology such that the family(Uγ | γ ∈ Γ) is a fundamental system of open neighborhoods of0. The argument in (ii) showsmore precisely thatKs is dense inKa for the valuation topology of(Ka, | · |Ka).

(iv) If ∆ ⊂ Γ is any subgroup, thenc.rk(Γ) ≤ c.rk(∆) + rk(Γ/∆) (cp. [11, Ch.VI,§10, n.2,Prop.3]).

(v) A subgroup∆ ⊂ Γ is convex if and only if there is an ordered group structure onΓ/∆such that the natural mapΓ→ Γ/∆ is order-preserving. Then the ordered group structure withthis property is unique.

(vi) If c.rk(Γ) = 1, we can find an order-preserving imbedding

ρ : (Γ, ·,≤) → (R,+,≤).

Indeed, pick an elementg ∈ Γ with g > 1. For everyh ∈ Γ, and every positive integern,there exists a largest integerk(n) such thatgk(n) < hn. Then(k(n)/n | n ∈ N) is a Cauchysequence and we letρ(h) := lim

n→∞k(n)/n. One verifies easily thatρ is an order-preserving

group homomorphism, and since the convex rank ofΓ equals one, it follows thatρ is injective.


5.1.5.1.22. There is an inclusion-reversing bijection between the set of convex subgroups ofthe value groupΓ of a valuation| · | and the set of prime ideals of its valuation ringK+.This bijection assigns to a convex subgroup∆ ⊂ Γ, the prime idealp∆ := x ∈ K+ | γ >|x| for everyγ ∈ ∆. Conversely, to a prime idealp, there corresponds the convex subgroup∆p := γ ∈ Γ | γ > |x| for all x ∈ p. Then, the value group of the valuation ringK+

p is(naturally isomorphic to)Γ/∆p. Furthermore,K+/p is a valuation ring of its field of fractions,and its value group is∆p.

5.1.5.1.23. Therank of a valuationis defined as the convex rank of its value group. It is clearfrom (5.1.5.1.22) that this is the same as the Krull dimension of the associated valuation ring.

5.1.5.1.24. For any field extensionF1 ⊂ F2, denote bytr.d(F2 : F1) the transcendence degreeof F2 over F1. Let E be a field extension of the valued fieldK, and | · |E : E× → ΓEan extension of the valuation| · |K : K× → ΓK of K to E. Let κ (resp. κ(E)) be theresidue field of the valuation ring of(K, | · |) (resp. of(E, | · |)). Then we have the inequality:rk(ΓE/ΓK) + tr.d(κ(E) : κ) ≤ tr.d(E : K) (cp. [11, Ch.VI,§10, n.3, Cor.1]).

5.1.5.1.25. The image ofK+ \ 0 in Γ is the monoidΓ+. The submonoids ofΓ+ are inbijective correspondence with the multiplicative subsetsof K+ \ 0 which contain(K+)×.The bijection is exhibited by the following ”short exact sequence” of monoids:

1→ (K+)× → K+ \ 0π→ Γ+ → 1.

Then, to a monoidM ⊂ Γ one assigns the multiplicative subsetπ−1(M).

5.1.5.1.26. Let us say that a submonoidN of a monoidM is convexif the following holds. Ifγ, δ ∈ M andγ · δ ∈ N , thenγ ∈ N andδ ∈ N . For every submonoidN there is a smallestconvex submonoidN con such thatN ⊂ N con. One deduces a natural bijection between convexsubmonoids ofΓ+ and prime ideals ofK+, by assigning on one hand, to a convex monoidM ,the idealp(M) := K+ \ π−1(M), and on the other hand, to a prime idealp, the convex monoidM(p) := π(K+ \ p).

5.1.5.1.27. The subsets of the formM \ N , whereN is a convex submonoid of the monoidM , are the first examples of ideals in a monoid. More generally,one says that a subsetI ⊂ Mis an ideal of M , if I ·M ⊂ I. Then we say thatI is aprime idealif I is an ideal such that,for everyx, y ∈ M with x · y ∈ I, we have eitherx ∈ I or y ∈ I. Equivalently, an idealI isa prime ideal if and only ifM \ I is a submonoid; in this caseM \ I is necessarily a convexsubmonoid. For a monoidM , let us denote bySpec(M) the set of all the prime ideals ofM .Taking into account (5.1.5.1.22), we derive bijections

Spec(Γ)∼→ Spec(K+)

∼→ Spec(Γ+) : ∆ 7→ p∆ 7→ π(p∆) = Γ+ \∆+.

Furthermore, the bijectionSpec(K+)∼→ Spec(Γ+) extends to an inclusion-preserving bijection

between the ideals ofK+ and the ideals ofΓ+.In the sequel, it will be sometimes convenient to study a monoid via the system of its finitely

generated submonoids. In preparation for this, we want to delve a little further into the theoryof general commutative monoids.

Definition 5.1.28. LetM be a commutative monoid.

(i) We say thatM is integral if we havea = b, whenevera, b, c ∈ M anda · c = b · c.(ii) We say thatM is free if it isomorphic toN(I) for some index setI. In this case, a minimal

set of generators forM will be called abasis.


5.1.5.1.29. LetMnd be the category of commutative monoids. The natural forgetful functorZ-Mod → Mnd admits a left adjoint functorM 7→ Mgp. Given a monoidM , the abeliangroupMgp can be realized as the set of equivalence classes of pairs(a, b) ∈ M ×M , where(a, b) ∼ (a′, b′) if there existsc ∈ M such thata · b′ · c = a′ · b · c; the addition is definedtermwise, and the unit of the adjunction is the mapφ : M → Mgp : a 7→ (a, 1) for everya ∈M . It is easy to see thatφ is injective if and only ifM is integral.

5.1.5.1.30. The categoryMnd admits arbitrary limits and colimits. In particular, it admitsdirect sums. The functorM 7→Mgp commutes with limits and colimits.

Theorem 5.1.31.Let∆ be an ordered abelian group,N ⊂ ∆+ a finitely generated submonoid.Then there exists a free finitely generated submonoidN ′ ⊂ ∆+ such thatN ⊂ N ′.

Proof. SinceN is a submonoid of a group, it is integral, soN ⊂ Ngp. The group homomor-phismNgp ⊂ ∆ induced by the imbeddingN ⊂ ∆ is injective as well. The verification isstraightforward, using the description ofNgp in (5.1.5.1.29). ThenNgp inherits a structure ofordered group from∆, and we can replace∆ byNgp, thereby reducing to the case where∆ isfinitely generated andN spans∆. Thus, in our situation, the convex rankr of ∆ is finite; wewill argue by induction onr. Suppose then thatr = 1. In this case we will argue by inductionon the rankn of ∆. If n = 1, then one has only to observe thatZ+ is a free monoid. Supposenext thatn = 2; in this case, letg1, g2 ∈ ∆ be a basis. We can suppose thatg1 < g2 < 1;indeed, ifg1 > 1, we can replace it byg−11 ; then, sincer = 1, we can find an integerk such thatg′2 := g2·gk1 < 1 andg′2 > g1; clearlyg1, g′2 is still a basis of∆. We define inductively a sequenceof elementsgi ∈ ∆+, for everyi > 2, in the following way. Suppose thati > 2 and that the ele-mentsg3 < g4 < ... < gi−1 have already been assigned; letki := supn ∈ N | gi−1 · g

−ni−2 ≤ 1;

notice that, since the convex rank of∆ equals1, we havek <∞. We setgi := gi−1 · g−kii−2 .

Claim 5.1.32. We havegNi · gNi+1 ⊂ gNi+1 · g

Ni+2 for everyi ≥ 1, and∆+ =

⋃i≥1(g

Ni · g

Ni+1).

Proof of the claim:The first assertion is obvious. We prove the second assertion. Let g ∈ ∆+;for everyi ≥ 1 we can writeg = gaii · g

bii+1 for uniqueai, bi ∈ Z. Notice thatai andbi cannot

both be negative. Suppose that eitherai+1 or bi+1 is not inN; we show that in this case

|ai+1|+ |bi+1| < |ai|+ |bi|.(5.1.33)

Indeed, we must have eitherai < 0 and bi > 0, or ai > 0 and bi < 0. However,ai+1 =ai · ki+1 + bi andbi+1 = ai; thus, ifai < 0, thenbi+1 < 0, and consequentlyai+1 > 0, whence

|ai+1| < |bi|(5.1.34)

and if ai > 0, thenai+1 < 0, so again (5.1.34) holds. From (5.1.33) it now follows thateventuallyai andbi become both positive.

SinceN is finitely generated, claim 5.1.32 shows thatN ⊂ gNi · gNi+1 for i > 0 sufficiently

large, so the claim follows in this case.Next, suppose that the convex rankr = 1 andn := rk(∆) > 2. Write∆ = H ⊕ G for two

subgroups such thatrk(H) = n− 1 andG = gZ for someg ∈ ∆.

Claim 5.1.35. For everyδ ∈ ∆+ \ 1 we can finda, b ∈ H such thatδ < a · g−1 < 1 andδ < b−1 · g < 1.

Proof of the claim:Let ρ : ∆ → R be an order-preserving imbedding as in example 5.1.21(vi);sincerk(H) > 1, it is easy to see thatρ(H) is dense inρ(∆). The claim is an immediateconsequence.

Let g1, ..., gk be a set of generators forN . For everyi ≤ k we can writegi = hi · gni foruniquehi ∈ H andni ∈ Z. Suppose thatni ≥ 0; it follows easily from claim 5.1.35 that thereexistsa, b ∈ H such thata < g < b andgi < (b−1 · g)ni < 1 (resp.gi < (a−1 · g)ni < 1) for


everyi such thatni ≥ 0 (resp.ni < 0). Then forni ≥ 0 seth′i := hi · bni and forni < 0 seth′i := hi · ani . Notice thath′i < 1. Seth0 := a · b−1 and letM be the submonoid ofH+ spannedby h0, h′1, ..., h

′k,; we can imbedM in a larger submonoidM ′ ⊂ H+ such that(M ′)gp = H.

Then, by inductive assumption, we can imbedM ′ in a free submonoidL ⊂ H+. Let l1, ..., ln−1be a basis forL.

We can writeh0 =∏t

i=1 lki for some integersk1, ..., kt ∈ 1, ..., n − 1. Notice thath0 ≤b−1 · g and letr < t be the largest integer such that

∏ri=1 lki > g · b−1; setl′ := g · b−1 ·

∏ri=1 l

−1ki

andl′′ := g−1 · b ·∏r+1

i=1 lki .

Claim5.1.36. The submonoidL′ generated byl1, ..., ln−1, l′, l′′ \ lkr+1 containsN .

Proof of the claim:Indeed, sincel′ · l′′ = lkr+1, it follows thatL ⊂ L′; moreover,g · b−1 =

l′ ·∏r

i=1 lki andg−1 · a = l′′ ·∏t

i=r+2 lki so g · b−1, g−1 · a ∈ L′. Now the claim follows byremarking thatgi = h′i · (g

−1 · a)−ni if ni < 0, andgi = h′i · (g · b−1)ni if ni ≥ 0.

Now, it is clear thatL ⊂ ∆+; since moreoverL spans∆ and is generated byn elements, itfollows thatL is a free monoid, so the proof is concluded in casec.rk(∆) = 1.

Finally, supposer > 1 and pick a convex subgroup0 6= ∆0 ( ∆; then the ordering on∆induces a unique ordering on∆/∆0 such that the projection mapπ : ∆ → ∆/∆0 is order-preserving. LetN0 := π(N). By induction,N0 can be imbedded into a finitely generatedfree submonoidF0 of (∆/∆0)

+. By lifting a minimal set of generators ofF0 to elementsf1, ..., fn ∈ ∆+, we obtain a free finitely generated monoidF ⊂ ∆+ with π(F ) = F0. Now,choose a finite setS of generators forN ; we can partitionS = S1 ∪ S2, whereS1 = S ∩ ∆0

andS2 = S \∆0. By construction, for everyx ∈ S2 there exist integerski,x ≥ 0 (i = 1, ..., n)such thatyx := x ·

∏ni=1 f

−ki,xi ∈ ∆0. Let g := maxyx | x ∈ S2; if g < 1, let ei := fi,

otherwise letei := fi · g for everyi ≤ n. Since∆0 is convex, we have in any case:ei < 1

for i ≤ n. Moreover, the elementszx := x ·∏n

i=1 e−ki,xi are contained in∆+

0 . By induction,the submonoid of∆+

0 generated byS1 ∪ zx | x ∈ S2 is contained in a free finitely generatedmonoidF ′ ⊂ ∆+

0 . Using the convexity of∆0 one verifies easily thatN ′ := F · F ′ is a freemonoid. ClearlyN ⊂ N ′, so the assertion follows.

Remark 5.1.37. Another proof of theorem 5.1.31 can be found in [22, Th.2.2].Moreover, thistheorem can also be deduced from the resolution of singularities of toric varieties ([36, Ch.I,Th.11]).

5.2. Basic ramification theory. This section is a review of some basic ramification theory inthe setting of general valuation rings and their algebraic extensions.

5.2.5.2.1. Throughout this section we fix a valued field(K, | · |). Its valuation ring will bedenotedK+ and the residue field ofK+ will be denoted byκ. If (E, | · |E) is any valued fieldextension ofK, we will denote byE+ the valuation ring ofE, byκ(E) its residue field and byΓE its value group. Furthermore, we letKa be an algebraic closure ofK, andKs the separableclosure ofK contained inKa.

5.2.5.2.2. LetE ⊂ Ka be a finite extension ofK. LetW be the integral closure ofK+ in E;by remark 5.1.12(iv), to every maximal idealp of W we can associate a valuation| · |p : E× →Γp extending|·|, and (up to isomorphisms of value groups) every extension of|·| toE is obtainedin this way. Setκ(p) := W/p; it is known that

∑p∈Max(W )[Γp : Γ] · [κ(p) : κ] ≤ [E : K] (cp.

[11, Ch.VI §8, n.3 Th.1]).


5.2.5.2.3. Suppose now thatE is a Galois extension ofK. ThenGal(E/K) acts transitivelyon Max(W ). For a givenp ∈ Max(W ), thedecomposition subgroupDp ⊂ Gal(E/K) of pis the stabilizer ofp. Thenκ(p) is a normal extension ofκ and the natural morphismDp →Aut(κ(p)/κ) is surjective; its kernelIp is theinertia subgroupatp (cp. [11, Ch.V,§2, n.2, Th.2]for the case of a finite Galois extension; the general case is obtained by passage to the limit overthe family of finite Galois extensions ofK contained inE).

5.2.5.2.4. If nowE is a finite Galois extension ofK, then it follows easily from (5.2.5.2.2)and (5.2.5.2.3) that the integers[Γp : Γ] and[κ(p) : κ] are independent ofp, and therefore, ifWadmitsn maximal ideals, we have :n · [Γp : Γ] · [κ(p) : κ] ≤ [E : K].

Lemma 5.2.5. Let K+sh be a strict henselization ofK+; thenK+sh is a valuation ring andΓK+sh = Γ.

Proof. It was shown in remark 5.1.12(vi) thatK+sh is a valuation ring. To show the secondassertion, letR be more generally any integrally closed local domain; the (strict) henselizationofR can be constructed as follows (cp. [45, Ch.X,§2, Th.2]). LetF := Frac(R),F s a separableclosure ofF , p any maximal ideal of the integral closureW of R in F s, D andI respectivelythe decomposition and inertia subgroups ofp; let WD (resp.W I) be the subring of elementsof W fixed byD (resp. byI) and setpD := WD ∩ p, pI := W I ∩ p. Then the localizationRh := (WD)pD (resp. Rsh := (W I)pI ) is a henselization (resp. strict henselization) ofR.Now, let us makeR := K+, soF := K andF s := Ks; let E ⊂ Ks be any finite Galoisextension ofK; WE := W ∩ E is the integral closure ofK+ in E; setDE := D ∩Gal(E/K),IE := I ∩ Gal(E/K), E ′ := EDE , E ′′ := EIE . Let p′ := p ∩ E ′; it then follows from [47,Ch.VI,§12, Th.23] that[Γp′ : Γ] = 1. Clearly the value groupΓKh of K+h is the filtered unionof all suchΓp′, so we deduceΓKh = Γ. Therefore, in order to prove the lemma, we can assumethatK = Kh. In this caseGal(E/K) coincides with the decomposition subgroup ofp′ andIE is a normal subgroup ofGal(K/E) such that[Gal(K/E) : IE ] equals the cardinalitynof Aut(κ(E)/κ). By the definition ofIE it follows that the natural map :Aut(κ(E)/κ) →Aut(κ(E ′′)/κ) is an isomorphism. We derive :[E ′′ : K] = n ≤ [κ(E ′′) : κ]; then from(5.2.5.2.4) we obtainΓE′′ = Γ and the claim follows.

5.2.5.2.6. We suppose now thatK+ is ahenselianvaluation ring, with value groupΓ. Then,on any algebraic extensionE ⊂ Ka of K, there is a unique valuation| · |E extending| · |, andthus a unique inertia subgroup, which we denote simply byI. By remark 5.1.12(iv),E+ is theintegral closure ofK+ in E.

Remark 5.2.7. (i) In the situation of (5.2.5.2.6), the inequality of (5.2.5.2.2) simplifies to :

[κ(E) : κ] · [ΓE : ΓK ] ≤ [E : K].

(ii) Sometimes this inequality is actually an equality; this is for instance the case when thevaluation ofK is discrete and the extensionK ⊂ E is finite and separable (cp. [11, Ch.VI,§8,n.5, Cor.1]).

(iii) However, even when the valuation ofK is discrete, it may happen that the inequality (i)is strict, ifE is inseparable overK. As an example, letκ be a perfect field of positive character-istic, and choose a power seriesf(T ) ∈ κ[[T ]] which is transcendental over the subfieldκ[T ].EndowF := Frac(κ(T 1/p, f(T ))) with theT -adic valuation, and letK be the henselization ofF . Then the residue field ofK is κ and the valuation ofK is discrete. LetE := K[f(T )1/p].Then[E : K] = p, ΓE = ΓK andκ(E) = κ.


5.2.5.2.8. For a fieldF , we denote byµ(F ) the torsion subgroup ofF×. Let E be a finiteGalois extension ofK (with K+ still henselian). One defines a pairing

I × (ΓE/ΓK)→ µ(κ(E))(5.2.9)

in the following way. For(σ, γ) ∈ I × ΓE, let x ∈ E× such that|x| = γ; then let(σ, γ) 7→σ(x)/x ( mod mE). One verifies easily that this definition is independent of the choice ofx;moreover, ifx ∈ K×, thenσ acts trivially onx, so the definition is seen to depend only on theclass ofγ in ΓE/ΓK .

5.2.5.2.10. Suppose furthermore thatκ is separably closed. Then the inertia subgroup coin-cides with the Galois groupGal(E/K) and moreoverµ(κ(E)) = µ(κ). The pairing (5.2.9)induces a group homomorphism

Gal(E/K)→ HomZ(ΓE/ΓK ,µ(κ)).(5.2.11)

Let p := char(κ). For a groupG, let us denote byG(p) the maximal abelian quotient ofG thatdoes not containp-torsion.

Proposition 5.2.12.Under the assumptions of(5.2.5.2.10), the map(5.2.11)is surjective andits kernel is ap-group.

Proof. Let n := [E : K]. Notice thatµ(κ) does not containp-torsion, hence every homomor-phismΓE/ΓK → µ(κ) factors through(ΓE/ΓK)(p). Let m be the order of(ΓE/ΓK)(p). Letus recall the definition of the Kummer pairing: one takes the Galois cohomology of the exactsequence ofGal(Ks/K)-modules

1→ µm → (Ks)×(−)m

−→ (Ks)× → 1

and applies Hilbert90, to derive an isomorphismK×/(K×)m ≃ H1(Gal(Ks/K),µm). Now,since(m, p) = 1 andκ is separably closed, the equationXm = 1 admitsm distinct solutionsin κ. SinceK+ is henselian, these solutions lift to roots of1 in K, i.e., µm ⊂ K×, whenceH1(Gal(Ks/K),µm) ≃ Homcont(Gal(Ks/K),µm). By working out the identifications, onechecks easily that the resulting group isomorphism

K×/(K×)m ≃ Homcont(Gal(Ks/K),µm)

can be described as follows. To a givena ∈ K×, we assign the group homomorphism

Gal(Ks/K)→ µm : σ 7→ σ(a1/m)/a1/m.

Notice as well that, sinceκ is separably closed, more generally every equation of the formXm = u admitsm distinct solutions inκ, providedu 6= 0; again by the henselian property wededuce that every unit ofK+ is anm-th power inK×; thereforeK×/(K×)m ≃ ΓK/mΓK .

Dualizing, we obtain an isomorphism

HomZ(Γ/mΓ,µm) ≃ HomZ(Homcont(Gal(Ks/K),µm),µm).

However,

Homcont(Gal(Ks/K),µm) = colimH⊂Gal(Ks/K)

HomZ(Gal(Ks/K)/H,µm)

whereH runs over the cofiltered system of open normal subgroups ofGal(Ks/K) such thatGal(Ks/K)/H is abelian with exponent dividingm. It follows that

HomZ(Homcont(Gal(Ks/K)/H,µm),µm) ≃ limH⊂Gal(Ks/K)

Gal(Ks/K)/H


where the right-hand side is a quotient ofGal(Ks/K)(p). Hence, we have obtained a surjectivegroup homomorphism

Gal(Ks/K)→ HomZ(m−1ΓK/ΓK ,µm)

∼→ HomZ(m

−1ΓK/ΓK ,µ(κ)).(5.2.13)

(SinceΓK is torsion-free, we can identify naturallyΓK/mΓK to the subgroupm−1ΓK/ΓK ⊂(ΓK ⊗Z Q)/ΓK). Let j : ΓE/ΓK → m−1ΓK/ΓK be the inclusion map. One verifies directlyfrom the definitions, that the maps (5.2.11) and (5.2.13) fit into a commutative diagram

Gal(Ks/K) //

Gal(E/K)

HomZ(m

−1ΓK/ΓK ,µ(κ))ρ // HomZ(ΓE/ΓK ,µ(κ))

where the top map is the natural surjection, andρ := HomZ(j,µ(κ)). Finally, an easy applica-tion of Zorn’s lemma shows thatρ is surjective, and therefore, so is (5.2.11).

It remains to show that the kernelH of (5.2.11) is ap-group. Suppose thatσ ∈ H andneverthelessp does not divide the orderl of σ; then we claim that theK-linear mapφ : E → E

given byx 7→∑l−1

i=0 σi(x) is an isometry. Indeed,φ(x) = l · x +

∑l−1i=1(σ

i(x) − x); it sufficesthen to remark that|l · x| = |x| and|σi(x) − x| < |x|, sinceσi ∈ H for i = 0, ..., l − 1. Next,for everyx ∈ E we can write0 = σl(x)− x = φ(x− σ(x)); henceσ(x) = x, that is,σ is theneutral element ofGal(E/K), as asserted.

Corollary 5.2.14. Keep the assumptions of(5.2.5.2.10), and suppose moreover that(p, [E :K]) = 1. ThenΓE/ΓK ≃ HomZ(Gal(E/K),µ(K)). Moreover, ifΓE/ΓK ≃ Z/q1Z ⊕ ... ⊕Z/qkZ, then there exista1, ..., ak ∈ K withE = K[a

1/q11 , ..., a

1/qkk ].

Proof. To start out, since(p, [E : K]) = 1, proposition 5.2.12 tells us that the map (5.2.11)is an isomorphism. In particular,Gal(E/K) is abelian, and[ΓE : ΓK ] ≥ [E : K], whence[ΓE : ΓK ] = [E : K] by remark 5.2.7(ii). ThereforeGal(E/K) ≃ Z/q1Z ⊕ ... ⊕ Z/qkZ andE is a compositum of cyclic extensionsE1, ..., Ek of orderq1, ..., qk. It follows as well thatΓE/ΓK ≃ HomZ(Gal(E/K),µ(κ)), so the first assertion holds; furthermore the latter holdsalso for every extension ofK contained inE. We deduce :

Claim 5.2.15. The Galois correspondence establishes an inclusion preserving bijection betweenthe subgroups ofΓE containingΓK , and the subfields ofE containingK.

To prove the second assertion, we are thus reduced to the casewhereE is a cyclic extensionof prime power order, sayGal(E/K) ≃ Z/qZ, with (q, p) = 1. Let γ ∈ ΓE be an elementwhose class inΓE/ΓK is a generator; we can finda ∈ K such that|a| = γq. LetE ′ := K[a1/m]andF := E · E ′. SinceΓF is torsion-free, one sees easily that its subgroupsΓE andΓE′

coincide. However,F satisfies again the assumptions of the corollary, thereforeclaim 5.2.15applies toF , and yieldsE = E ′.

Definition 5.2.16. Let (K, | · |) be a valued field. We denote byK+sh be the strict henselisa-tion of K+ and setKsh := Frac(K+sh). Themaximal tame extensionKt of K in its separa-ble closureKs is the union of all the finite Galois extensionsE of Ksh insideKs, such that([E : Ksh], p) = 1. Notice that, by corollary 5.2.14, every such extension is abelian and thecompositum of two such extensions is again of the same type, so the family of all such finiteextension is filtered, and therefore their union is their colimit, so the definition makes sense.

5.2.5.2.17. SinceΓKsh = ΓK , one verifies easily from the foregoing that there is a naturalisomorphism of topological groupsGal(Kt/Ksh) ≃ HomZ(ΓK ⊗Z Z(p)/ΓK ,µ

(p)), whereµdenotes the group of roots of1 inKsh and where we endow the target with the profinite topology.


5.2.5.2.18. LetE ⊂ Ks be any separable extension ofK. Then it is easy to check thatEt = E ·Kt. Indeed, one knows thatEsh = E ·Ksh; then letF be a finite separable extensionof E such that([F : E], p) = 1. By taking roots of elements ofK we can find an extensionF ′

of K such that([F ′ : K], 1) = 1 and(ΓF/ΓK)(p) = (ΓF ′/ΓK)(p) and thenE · F ′ ·Ksh ⊃ F .

5.3. Algebraic extensions.In this section we return to almost rings: we suppose it is given avalued field(K, | · |), and then we will study exclusively the almost ring theory relative to thestandard setup attached toK+ (see (5.1.5.1.15)). For an extensionE of K, we will use thenotation of (5.2.5.2.1). Furthermore, we will denoteWE the integral closure ofK+ in E.

Lemma 5.3.1. LetR be a ring and0 → M1 → M2 → M3 → 0 a short exact sequence offinitely generated torsionR-modules, and suppose that theTor-dimension ofM3 is≤ 1. ThenF0(M2) = F0(M1) · F0(M3).

Proof. We can find epimorphismsφi : Rni → Mi for i ≤ 3, with n2 = n1 + n3, fitting into acommutative diagram with exact rows:

0 // Rn1 //

φ1

Rn2 //

φ2

Rn3 //

φ3

0

0 // M1// M2

// M3// 0.

LetNi := Ker(φi) (i ≤ 3). By snake lemma we have a short exact sequence:0→ N1 → N2π→

N3 → 0. Since the Tor-dimension of theM3 is≤ 1, it follows thatN3 is a flatR-module.

Claim5.3.2. Λn3+1R N3 = 0.

Proof of the claim:SinceN3 is flat, the antisymmetrizer operatorak : ΛkRN3 → N⊗k3 is injectivefor everyk ≥ 0 (cp. the proof of proposition 4.3.26). On the other hand,Λn3+1

R Rn3+1 = 0,thus it suffices to show that the natural mapj⊗k : N⊗k3 → (Rn3)⊗k is injective for everyk ≥ 0. This is clear fork = 0. Suppose that injectivity is known forj⊗k; we havej⊗k+1 =(1R⊗k ⊗R j) (j⊗k ⊗R 1N3). SinceN3 is flat, we conclude by induction onk.

Next recall that, for everyk ≥ 0 there are exact sequences

N1 ⊗R ΛkN2 → Λk+1N2π∧k+1

−→ Λk+1N3.(5.3.3)

(To show that such sequences are exact, one uses the universality of Λk+1R N3 for (k + 1)-linear

alternating maps toR-modules). From (5.3.3) and claim 5.3.2, a simple argument by inductionon k shows that the natural mapψ : Λn1

R N1 ⊗ Λn3R N2 → Λn2

R N2 is surjective. Finally, by

definition, we haveF0(Ni) = Im(ΛniRNi

j∧nii−→ Λni

RRni∼→ R). To conclude, it suffices therefore

to remark that the diagram:

Λn1R N1 ⊗ Λn3

R N2

1Λn1R

N1⊗π∧k+1

ψ // Λn2R N2

j∧n22

Λn1R N1 ⊗ Λn3

R N3// Λn2

R Rn2

(5.3.4)

commutes. We leave to the reader the task of verifying that the commutativity of (5.3.4) boilsdown to a well-known identity for determinants of matrices.

Remark 5.3.5. (i) Lemma 5.3.1 applies especially to a short exact sequenceof finitely pre-sented torsionK+-modules, since by lemma 5.1.14, any such module has homological dimen-sion≤ 1.


(ii) By the usual density arguments (cp. the proof of proposition 2.3.23), it then follows thatlemma 5.3.1 holds trueverbatim, even when we replaceR byK+ and theR-modulesM1,M2,M3 by uniformly almost finitely generated torsionK+a-modules.

Proposition 5.3.6. Suppose thatK+ is a valuation ring of rank one. LetE be a finite separableextension ofK. ThenW a

E andΩW aE/K

+a are uniformly almost finitely generatedK+a-moduleswhich admit the uniform bounds[E : K] and respectively[E : K]2. Moreover,W a

E is an almostprojectiveK+a-module.

Proof. In view of the presentation (2.5.26), the assertion forΩW aE/K

+a is an immediate conse-quence of the assertion forW a

E . The trace pairingtE/K : E × E → K is perfect sinceE isseparable overK. Let e1, ..., en be a basis of theK-vector spaceE ande∗1, ..., e

∗n the dual basis

under the trace morphism, so thattE/K(ei ⊗ e∗j ) = δij for everyi, j ≤ n. We can assume thatei ∈ WE and we can finda ∈ K+ \0 such thata · e∗i ∈ WE for everyi ≤ n. Letw ∈ WE ; wecan writew =

∑ni=1 ai · ei for someai ∈ K. We havetE/K(w ⊗ a · e∗j) ∈ K

+ for everyj ≤ n;on the other hand,tE/K(w ⊗ a · e∗j) = a · aj . Thus, if we letφ : Kn → E be the isomorphism(x1, ..., xn) 7→

∑ni=1 xi · ei, we see that

(K+)n ⊂ φ−1(WE) ⊂ a−1 · (K+)n.(5.3.7)

We can writeWE as the colimit of the familyW of all its finitely generatedK+-submodulescontaininge1, ..., en; if W0 ∈ W , thenW0 is a freeK+-module by remark 5.1.12(ii); then it isclear from (5.3.7) that the rank ofW0 must be equal ton. The proof follows straightforwardlyfrom the following:

Claim 5.3.8. Let ε ∈ m; there existsW0 ∈ W such thatε ·WE ⊂W0.

Proof of the claim: Indeed, suppose that this is not the case. Then we can find an infinitesequence of finitely generated submodules⊕ni=1ei · K

+ ⊂ W0 ⊂ W1 ⊂ W2 ⊂ ... ⊂ WE

such thatε ·Wi+1 * Wi for everyi ≥ 0. From (5.3.7) and lemma 5.3.1 it follows easily thatF0((K

+)n/a · (K+)n) ⊂ F0(Wk+1/W0) =∏k

i=0 F0(Wi+1/Wi) for everyk ≥ 0. However,F0(Wi+1/Wi) ⊂ AnnK+(Wi+1/Wi) ⊂ ε ·K+ for everyi ≥ 0. We deduce that|a|n < |ε|k foreveryk ≥ 0, which is absurd, since the valuation ofK has rank one.

5.3.5.3.9. Suppose that the valuation ring ofK has rank one. LetK ⊂ E ⊂ F be a towerof finite separable extensions. Letp ⊂ WE be any prime ideal; thenWE,p is a valuation ring(see remark 5.1.12(iii)), andWF,p is the integral closure ofWE,p in F . It then follows fromproposition 5.3.6 and remark 5.1.12(ii) thatW a

F,p is an almost finitely generated projectiveW aE,p-

module; we deduce thatW aF is an almost finitely generated projectiveW a

E-module, therefore wecan define the different ideal ofW a

F overW aE . To ease notation, we will denote it byDWF /WE

.If | · |F is a valuation ofF extending| · |, thenF+ = WF,p for some prime idealp ⊂ WF ;moreover, if| · |E is the restriction of| · |F toE, thenE+ =WE,q, whereq = p ∩WE. For thisreason, we are led to defineDF+/E+ := (DWF /WE

)p.

Lemma 5.3.10.LetK ⊂ E ⊂ F be a tower of finite separable extensions ofK. Then:(i) TheW a

E-module(W aE)∗ is invertible.

(ii) DWE/K+ ·DWF /WE= DWF /K+.

Proof. In view of proposition 4.1.27, (ii) follows from (i). We show(i): from proposition5.3.6 we can find, for everyε ∈ m, a finitely generatedK+-submoduleM ⊂ WE such thatε ·WE ⊂M . By remark 5.1.12(ii) it follows thatM is a freeK+-module, so the same holds forM∗ := HomK+(M,K+). The scalar multiplicationM∗ → M∗ : φ 7→ ε · φ factors through amapM∗ → W ∗

E, and if we letN be theWE-module generated by image of the latter map, thenε ·W ∗

E ⊂ N . Furthermore, for every prime idealp ⊂ WE , the localizationNp is a torsion-free


WE,p-module; sinceWE,p is a valuation ring, it follows thatNp is free of finite rank, again byremark 5.1.12(ii). Hence,N is a projectiveWE-module. In particular, this shows that(W a

E)∗ is

almost finitely generated projective as aW aE-module. To show that(W a

E)∗ is also invertible, it

will suffice to show that the rank ofN equals one. However, the rank ofN can be computed asdimE N ⊗WE

E. We haveN ⊗WEE = W ∗

E ⊗K+ K = HomK(E,K), so the assertion followsby comparing the dimensions of the two sides.

Proposition 5.3.11.Suppose thatK+ has rank one. LetK ⊂ E be a finite field extension suchthat l := [E : K] is a prime. Letp := char(κ). Suppose that either:

(a) l 6= p andK = Ksh, or(b) l = p andK = Kt, or(c) the valuation ofK is discrete and henselian, andE is separable overK, or(d) the valuation ofK is discrete and henselian,ΓE = ΓK andκ(E) = κ.

Then :

(i) In case(a), (b) or (d) holds, there existsx ∈ E \K such thatE+ is the filtered union of afamily of finiteK+-subalgebras of the formE+

i := K+[aix+ bi], (i ∈ N) whereai, bi ∈ Kare elements with|aix+ bi| ≤ 1.

(ii) In case(c) holds, there exists an elementx ∈ E+ such thatE+ = K+[x].(iii) Furthermore, ifE is a separable extension ofK, thenHj(LE+/K+) = 0 for everyj > 0.(iv) If E is an inseparable extension ofK, thenHj(LE+/K+) = 0 for everyj > 1, and

moreoverH1(LE+/K+) is a torsion-freeE+-module.

Proof. Let us first show how assertions (iii) and (iv) follow from (i)and (ii). Indeed, since thecotangent complex commutes with colimits of algebras, by (i) and (ii) we reduce to dealingwith an algebra of the formK+[w] for w ∈ E+. Such an algebra is a complete intersectionK+-algebra, quotient of the free algebraK+[X ] by the idealI ⊂ K+[X ] generated by the minimalpolynomialm(X) of w. In view of [34, Ch.III, Cor.3.2.7], one has a natural isomorphism inD(K+[w]-Mod)

LK+[w]/K+ ≃ (0→ I/I2δ→ ΩK+[X]/K+ ⊗K+ K+[w]→ 0).

If we identify ΩK+[X]/K+ ⊗K+ K+[w] to the rank one freeK+[w]-module generated bydX,thenδ can be given explicitly by the rule:f(X) 7→ f ′(w)dX, for everyf(X) ∈ (m(X)).However,E is separable overK if and only ifm′(w) 6= 0. It follows thatδ is injective if andonly if E is separable overK, which proves (iii). IfE is inseparable overK, thenδ vanishesidentically by the same token. This shows (iv).

We prove (ii). Since the valuation is discrete, we must have either e := [ΓE : Γ] = l orf := [κ(E) : κ] = l (see remark 5.2.7(ii)). Ife = l, then pick any uniformizera ∈ E; everyelement ofE can be written as a sum

∑l−1i=0 xi · a

i with xi ∈ K for every i < l. Then it iseasy to see that such a sum is inE+ if and onlyxi ∈ K+ for every i < l. In other words,E+ = K+[a]. In casef = l, we can writeκ(E) = κ[u] for some unitu ∈ (E+)×; moreover,mE = m · E+; thenK+[u] + m · E+ = E+; since in this caseE+ is a finiteK+-module, wededuceE+ = K+[u] by Nakayama’s lemma.

We prove (i). Suppose that (a) holds; then by corollary 5.2.14 it follows thatΓE/ΓK ≃ Z/lZandE = K[a1/l] for somea ∈ K. Hence:

|ai/l| /∈ Γ for everyi = 1, ..., l − 1.(5.3.12)

We can suppose that the valuation ofK is not discrete, otherwise we fall back on case (c); then,for everyε ∈ m, there existsbε ∈ K such that|ε| < |blε · a| < 1. Let x0, ..., xl−1 ∈ K and setw :=

∑l−1i=0 xi · a

i/l. Clearly every element ofE can be written in this form. From (5.3.12) wederive that the values|xi · ai/l| such thatxi 6= 0 are all distinct. Hence,|w| = max

0≤i<l|xi · a

i/l|.


Suppose now thatw ∈ E+; it follows that |xi · ai/l| ≤ 1 for i = 0, ..., l − 1, and in fact|xi · ai/l| < 1 for i 6= 0. Let ε ∈ m such that|εl−1| > |xi · ai/l| for everyi 6= 0. A simplecalculation shows that|xi · b−iε | < 1 for everyi 6= 0, in other words,w ∈ K+[bε · a

1/l], whichproves the claim in this case.

In order to deal with cases (b) and (d) we need some preparation. Let x ∈ E \ K be anyelement, and set:

ρ(x) := infa∈K|x− a| ∈ Γ∧E.

We consider case (b). Notice that the hypothesisK = Kt implies that the valuation ofKis not discrete. For anyy ∈ E we can writey = f(x) for somef(X) := b0 + b1X + ... +bdX

d ∈ K[X ] with d := deg f(X) < p. The degree of the minimal Galois extensionF of Kcontaining all the roots off(X) dividesd!, henceF ⊂ Kt = K. In other words, we can writey = ak ·

∏di=1(x− αi) for someα1, ..., αd ∈ K.

We distinguish two cases: first, suppose that there existsa ∈ K with |x − a| = ρ(x).Replacingx by x − a we may achieve that|x| ≤ |x − a| for everya ∈ K. Then the constantsequence(an := 0 | n ∈ N) fulfills the condition of lemma 5.1.9. Thus, ify = f(x) as above isin E+, we must have|f(X)|(0,ρ(x)) ≤ 1; in other terms:

|bi| · ρ(x)i ≤ 1 for everyi ≤ d.(5.3.13)

Now, if ρ(x) ∈ ΓK , we can findc ∈ K such thatx0 := x · c still generatesE and |x0| = 1,whence|bi/ci| ≤ 1 for everyi ≤ 1; however,y = b0 + (b1/c) · x0+ (b2/c

2)x20 + ...+ (bd/cd)xd,

thusy ∈ K+[x0], so in this case,E+ itself is one of theE+i .

In caseρ(x) /∈ ΓK , since anywayΓK is of rank one and not discrete, we can find a sequenceof elementsc1, c2, ... ∈ K such that, lettingxi := x · ci, we have

|xj − a| ≥ |xj | for everya ∈ K, j ∈ N; |xj| < 1 and |xj | → 1.

Claim 5.3.14. If x /∈ ΓK , then|xl| /∈ ΓK for every0 < l < p.

Proof of the claim:Indeed, suppose that|xl| ∈ ΓK for some0 < l < p; sinceΓK is l-divisible,we can multiplyx by somea ∈ K to have|xl| = 1, therefore|x| = 1, a contradiction.

From (5.3.13) and claim 5.3.14 we deduce that actually|bi| · ρ(x)i < 1 wheneveri > 0. Itfollows that, forj sufficiently large, we will have1 > |xij | > |bi| ·ρ(x)

i for everyi > 0. Writingy = b0+(b1/cj)xj+(b2/c

2j)x

2j + ...+(bd/c

dj )x

dj we deducey ∈ K+[xj ], therefore the sequence

of K+-subalgebraK+[ci · x] will do in this case.Finally we have to consider the case where the infimumρ(x) is not attained for anyx ∈ E.

In this case, since the valuation is not discrete and of rank1, we can find, for everyx ∈ E, asequence of elementsa0, a1, a2, ... ∈ K such that

γj := |x− aj | → ρ(x).(5.3.15)

In particular, forj sufficiently large we will have|x| > |x − aj |, therefore|x| = |aj|. Thisshows:

ΓE = ΓK .(5.3.16)

Now, pickx ∈ E \K and any sequence of elementsai ∈ K such that (5.3.15) holds; it is clearthat(ai | i ∈ N) fulfills the condition of lemma 5.1.9. Consequently

|y| = |f(X)|(aj ,γj) for every sufficiently largej.(5.3.17)

Let f(X) = b0,j + b1,j(X − aj) + ...+ bd,j(X − aj)d. (5.3.17) says that|bi,j| · γij ≤ 1 wheneverj is sufficiently large. However, from (5.3.16) we know thatγj ∈ ΓK . Pick cj ∈ K such that|cj | = γ−1j and setxj := cj(x − aj). It follows that|bi,j/cij| ≤ 1 andy = b0,j + (b1,j/c1,j)xj +


... + (bd,j/cdj )x

dj . Hencey ∈ K+[xj ]. It is then easy to verify that the family of all suchK+-

subalgebras is filtered by inclusion, and thus conclude the proof of case (b).At last, we turn to case (d). Notice that, by remark 5.2.7(ii), this case can occur only ifE is

inseparable overK, and thenl = p. Let x ∈ E \ K; let a ∈ m be a uniformizer; for givenn ∈ N, suppose thatbn ∈ K has been found such that|x − bn| ≤ |an|. Sinceκ(E) = κ, wecan find an elementc ∈ K+ such thatc ≡ (x − bn)/a

n (mod m). Setbn+1 := bn + c · an;then|x − bn+1| ≤ |a

n+1|. This shows thatρ(x) = 0, and the resulting sequence(bn | n ∈ N)converges tox in them-adic topology. Lety ∈ E; we can writey = f(x) for a polynomialf(X) ∈ K[X ] of degreed < p. LetF be the minimal field extension ofK that contains all theroots off(X). Notice that[F : K] dividesd!, henceF is separable overK, and[E ·F : F ] = p.Let f(X) = c ·

∏di=0(X−αi) be the factorization off(X) in F [X ]. By lemma 5.1.9 we deduce

that, for every sufficiently largen ∈ N we have:|y| = |f(X)|(bn,|x−bn|), where| · |(bn,|x−bn|)is the Gauss valuation onF (X). One then argues as in the proof of case (b), to show thaty ∈ E+

n := K+[cn(x− bn)], with cn ∈ K such that|cn(x− bn)| = 1. Again, it is easy to verifythatE+

i ⊂ E+i+1 for everyi ∈ N, so the proof is complete.

Corollary 5.3.18. LetE be a finite field extension ofK of prime degreel.

(i) If E satisfies condition(a)of proposition 5.3.11, and the valuation ofK is not discrete (butstill of rank one), thenΩE+/K+ = 0, LE+/K+ ≃ 0 andDE+/K+ = E+a.

(ii) If E satisfies condition(c) of proposition 5.3.11, then we have :F0(ΩE+/K+) = DE+/K+

andHi(LE+/K+) = 0 for i > 0.

Proof. (i): Since condition (a) holds, proposition 5.3.11 and its proof show that there existsa ∈K such thatE+ is the increasing union of allK+-subalgebras of the formE+

b := K+[b · a1/l],whereb ∈ K+ ranges over all elements such that|bl · a| < 1. Consequently,ΩE+/K+ =colim

bΩE+

b /K+, andLE+/K+ = colim

bLE+

b /K+. Then, again from proposition 5.3.11 it follows

thatHj(LE+/K+) = 0 for every j > 0. Hence, in order to show the first two assertions, itsuffices to show that the filtered system of theΩE+

b /K+ is essentially zero. However, theE+

b -

moduleΩE+b /K

+ is generated byωb := d(b · a1/l), and clearlyl · (bl · a)(l−1)/l · ωb = 0. Since

(l, p) = 1, it follows that(bl · a)(l−1)/l · ωb = 0. On the other hand, for|b| < |c| we can write:ωb = b·c−1·ωc. Therefore, the image ofωb inΩE+

c /K+ vanishes, whenever|b·c−1| < |cl·a|(l−1)/l,

i.e., whenever|b · a1/l| < |cl · a| < 1. Since the valuation ofK is of rank one and not discrete,such ac can always be found. To show the last stated equality, let us recall the following generalfact (for whose proof we refer to [45, Ch.VII,§1]).

Claim5.3.19. Suppose thatE = K[w] for somew ∈ E, and letf(X) ∈ K[X ] be its minimalpolynomial; the elements1, w, w2, ..., wl−1 form a basis of theK-vector spaceE. Let e∗1, ..., e

∗n

be the corresponding dual basis under the trace pairing; then the basesS := e∗1, ..., e∗n and

S ′ := wl−1/f ′(w), wl−2/f ′(w), ..., 1/f ′(w) span the sameE+-submodule ofE.

Let us takew = b · a1/l for someb ∈ V such that|bl · a| < 1. It follows from claim 5.3.19that(DE+/K+)−1 ⊂ f ′(w)−1 ·E+a, whencef ′(w) ∈ DE+/K+∗. However,f ′(w) = l ·wl−1, andfrom the definition ofw we see that|f ′(w)| can be made arbitrarily close to1, by choosing|b|closer and closer to|a|1/l.

(ii): the claim about the cotangent complex is just a restatement of proposition 5.3.11(iii),(iv).By proposition 5.3.11(ii) we can writeVE = K+[w] for somew ∈ E+. Let f(x) ∈ K+[X ]be the minimal polynomial ofw. Claim 5.3.19 implies thatDE+/K+ = (f ′(w)); a standardcalculation yieldsΩE+/K+ ≃ E+/(f ′(w)), so the assertion holds.


Theorem 5.3.20.Let (E, | · |E) be a finite separable valued field extension of(K, | · |) andsuppose thatK+ has rank one. ThenF0(ΩE+a/K+a) = DE+/K+ andHi(LE+/K+) = 0 fori > 0.

Proof. We begin with a few reductions:

Claim 5.3.21. We can assume thatE is a Galois extension ofK.

Proof of the claim: Indeed, let(L, | · |L) be a Galois valued field extension ofK extending(E, | · |E). We obtain by transitivity ([34, II.2.1.2]) a distinguished triangle

σ−1LL+/E+ → LE+/K+ ⊗E+ L+ → LL+/K+ → LL+/E+ .(5.3.22)

Suppose that the theorem is already known for the Galois extensionsK ⊂ L andE ⊂ L. Then(5.3.22) implies thatHi(LE+/K+) = 0 for i > 0 and moreover provides a short exact sequence

0→ ΩE/K+ ⊗E+ L+ → ΩL+/K+ → ΩL+/E+ → 0.

However, on one hand, by lemma 5.3.10(ii) the different is multiplicative in towers of exten-sions, and the other hand, the Fitting idealF0 is multiplicative for short exact sequences, byvirtue of remark (5.3.5)(ii), so the claim follows.

Claim 5.3.23. We can assume thatK+ is strictly henselian.

Proof of the claim:Indeed, letK+sh be the strict henselisation ofK+ andKsh := Frac(K+sh).It is known thatK+sh is an ind-etale extension ofK+, thereforeE+⊗K+K+sh is a reduced nor-mal semilocal integral and flatK+sh-algebra, whence a product of reduced normal local integraland flatK+sh-algebrasW1, ...,Wk. Each suchWi is necessarily the integral closure ofK+sh inEi := Frac(Wi). It follows thatLE+/K+ ⊗K+ K+sh ≃ LE+⊗K+K+sh/K+sh ≃ ⊕ki=1LWi⊗K+K+sh.Furthermore:DE+/K+ ⊗K+a (K+sh)a ≃ ⊕ki=1DE+

i /K+sh and similarly for the modules of dif-

ferentials. We remark as well that the formation of Fitting ideals commutes with arbitrary basechanges. In conclusion, it is clear that the assertions of the theorem hold for the extensionK ⊂ E if and only if they hold for each extensionKsh ⊂ Ei.

Claim 5.3.24. SupposeK = Ksh. We can assume thatGal(E/K) is ap-group.

Proof of the claim:Indeed, letP be the kernel of (5.2.11). By proposition 5.2.12,P is p-group;let L be the fixed field ofP . ThenL ⊂ Kt and, by virtue of corollary 5.2.14, we see thatLadmits a chain of subextensionsK := L0 ⊂ L1 ⊂ ... ⊂ Lk := L such that eachLi ⊂ Li+1

satisfies either condition (a) or (c) of proposition 5.3.11.Then, by corollary 5.3.18 it followsthat the assertions of the theorem are already known for the extensionsLi ⊂ Li+1. From here,using transitivity of the cotangent complex and multiplicativity of the different in towers ofextensions, and of the Fitting ideals for short exact sequences, one shows that the assertionshold also for the extensionK ⊂ L (cp. the proof of claim 5.3.21). Now, if the assertions areknown to hold as well for the extensionL ⊂ E, again the same argument proves them forK ⊂ E.

Claim 5.3.25. The theorem holds if the valuation ofK is discrete.

Proof of the claim:By claim 5.3.24, we can suppose thatGal(E/K) is a p-group. Hence,we can find a sequence of subextensionsE0 := K ⊂ E1 ⊂ E2 ⊂ ... ⊂ En := E with[Ei+1 : Ei] = p, for everyi = 0, ..., n− 1. Arguing like in the proof of claim 5.3.24 we see thatit suffices to prove the claim for each of the extensionsEi ⊂ Ei+1. In this case we are left todealing with an extensionK ⊂ E of degreep, which is taken care of by corollary 5.3.18(ii).

Claim 5.3.26. SupposeK = Ksh, thatGal(E : K) is a p-group and that the valuation ofKis not discrete. LetL be a finite Galois extension ofK such that([L : K], p) = 1. Then thenatural mapE+ ⊗K+ L+ → (E · L)+ is an isomorphism.


Proof of the claim:By corollary 5.2.14 we know thatL admits a tower of subextensions of theform K := L0 ⊂ L1 ⊂ ... ⊂ Lk := L, such that, for eachi ≤ k we haveLi+1 = Li[a

1/l] forsomea ∈ Li and some primel 6= p. By induction oni, we can then reduce to the case whereL = K[a1/l] for somea ∈ K and a primel 6= p. Under the above assumptions, we must haveE∩L = K, hencea /∈ E. ThenE ·L = E[a1/l] and by proposition 5.3.11 and its proof,(E ·L)+

is the filtered union of all its subalgebras of the formE+[b · a1/l], whereb ∈ E ranges over allthe elements such that|bm · a| < 1. However, since the valuation ofK is not discrete and hasrank one,ΓK is dense inΓE, and consequently the subfamily consisting of theE+[b · a1/l] withb ∈ K is cofinal. Finally, forb ∈ K we haveE+[b · a1/l] ≃ E+ ⊗K+ K+[b · a1/l]. By takingcolimits, it follows that(E · L)+ ≃ E+ ⊗K+ L+.

Claim5.3.27. We can assume thatK is equal toKt.

Proof of the claim:By claim 5.3.23 we can and do assume thatK = Ksh, in which caseKt

is the filtered union of all the finite Galois extensionL of K such that([L : K], p) = 1. ThenKt+ =

⋃L L

+ and(E ·Kt)+ =⋃L(E ·L)

+, whereL ranges over all such extensions. By claim5.3.24 we can also assume thatGal(E/K) is ap-group, in which case, by claim 5.3.26, we haveE+⊗K+L+ ∼

→ (E ·L)+ for everyL as above. Taking colimit, we getE+⊗K+Kt+ ∼→ (E ·Kt)+.

SinceKt+ is faithfully flat overK+, this shows that, in order to prove the theorem, we canreplaceK byKt; however, by (5.2.5.2.18) we have(Kt)t = Kt, whence the claim.

After this preparation, we are ready to finish the proof of thetheorem. We are reduced toconsidering a Galois extensionE of K = Kt such thatGal(E/K) is a p-group; moreover,we can assume that the valuation ofK is not discrete. Then, arguing as in the proof of claim5.3.25, we can further reduce to dealing with an extensionK ⊂ E of degreep; furthermore,the conditionK = Kt still holds, by virtue of (5.2.5.2.18). In this situation, condition (b) ofproposition 5.3.11 is fulfilled, henceHj(LE+/K+) = 0 for j > 0, by proposition 5.3.11(iii).It remains to show the identityF0(ΩEa+/K+a) = DE+/K+ . By proposition 5.3.11(i), thereexistsx ∈ E such thatE+ is the filtered union of a family of finiteK+-subalgebrasE+

i :=K+[aix + bi] (i ∈ N) of E+. Let f(X) ∈ K+[X ] be the minimal polynomial ofx. Byconstruction ofE+

i , it is clear that they form a Cauchy net inIK+a(E+a) converging toE+a.It then follows from lemma 2.5.25, that the netΩE+

i /K+ ⊗E+

iE+ | i ∈ N converges to

ΩE+/K+ in M (E+). In particular,F0(ΩE+/K+) = limi→∞

F0(ΩE+i /K

+ ⊗E+iE+). The minimal

polynomial ofaixi+ bi is fi(X) := f(a−1i X − bi), therefore:ΩE+i /K

+ = E+i /(f

′i(aixi+ bi)) =

E+i /(a

−1i f ′(x)). Consequently,F0(ΩE+/K+) = lim

i→∞(a−1i f ′(x)). On the other hand, claim

5.3.19 yields:DE+i /K

+ = (a−1i f ′(x)) for every i ∈ N. Then the claim follows from lemma4.1.30.

The final theorem of this section completes and extends theorem 5.3.20 to include valuationsof arbitrary rank.

Theorem 5.3.28.Let (K, | · |) be any valued field and(E, | · |E) any algebraic valued fieldextension of(K, | · |). We have :

(i) Hi(LE+/K+) = 0 for i > 1 andH1(LE+/K+) is a torsion-freeE+-module.(ii) If moreover,E is a separable extension ofK, thenHi(LE+/K+) = 0 for i > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, suppose thatE is separable overK. ThenLE/K ≃ 0. However, by (i), the natural mapH1(LE+/K+)→ H1(LE+/K+)⊗K+ K ≃H1(LE/K) is injective, so the assertion follows.

In order to prove (i), we reduce easily to the case of a finite algebraic extension. Let us writeK as the filtered union of its subfieldsLα that are finitely generated over the prime field. Foreach suchLα, letKα := (Lα)

a ∩K andEα := (Lα)a ∩E. ThenEα is a finite extension ofKα


andK is the filtered union of theKα. It follows easily that we can replace the extensionK ⊂ Eby the extensionKα ⊂ Eα, thereby reducing to the case where the transcendence degree ofKover its prime field is finite. In this situation, the rankr of K is finite (cp. (5.1.5.1.24)). Weargue by induction onr. Suppose first thatr = 1. We can split into a tower of extensionsK ⊂ Ks ∩ E ⊂ E; then, by using transitivity (cp. the proof of claim 5.3.21), we reduceeasily to prove the assertion for the subextensionsK ⊂ Ks andKs ∩ E ⊂ E. However, thefirst case is already covered by theorem 5.3.20, so we can assume thatE is purely inseparableoverK. In this case, we can further splitE into a tower of subextensions of degree equalto p; thus we reduce to the case where[E : K] = p. We apply transitivity to the towerK ⊂ Kt ⊂ E · Kt = Et: by proposition 5.3.11(iv) we know thatHi(LEt+/Kt+) vanishesfor i > 1 and is torsion-free fori = 1; by theorem 5.3.20, we haveHi(LKt+/K+) = 0 fori > 0, thereforeHi(LEt+/K+) vanishes fori > 1 and is torsion-free fori = 1. Next we applytransitivity to the towerK ⊂ E ⊂ Et : by theorem 5.3.20 we haveHi(LEt+/E+) = 0 for i > 0,and the claim follows easily.

Next suppose thatr > 1, and that the theorem is already known for ranks< r. Arguing as inthe proof of claim 5.3.23, we can even reduce to the case whereK+ is henselian, and thenE+

is the integral closure ofK+ in E. Letpr := (0) ⊂ pr−1 ⊂ ... ⊂ p0 be the chain of prime idealsof K+, and for everyi ≤ r let qi be the unique prime ideal ofE+ lying overpi. The valuationring E+

q1has rankr − 1, thus, by inductive assumption, the desired assertions areknown for

the extensionK+p1⊂ E+

q1. It suffices therefore to show thatHi(LE+/K+) ⊂ Hi(LE+

q1/K+

p1) for

everyi ≥ 0. Pick a ∈ p0 \ p1. ThenK+p1

= K+[a−1] andE+q1

= E+[a−1] andLE+q1/K+

p1=

LE+/K+ ⊗K+ K+[a−1]. Hence, we are reduced to show that multiplication bya is injective onthe homology ofLE+/K+. LetR := K+/aK+ andRE := E+ ⊗K+ R. We have a short exactsequence0 → K+ a

→ K+ → R → 0, therefore, after tensoring byLE+/K+ , a distinguishedtriangle:

LE+/K+a→ LE+/K+ → LE+/K+

L

⊗K+ R→ σLE+/K+ .

On the other hand, according to remark 5.1.12(ii),E+ is flat overK+, thereforeLE+/K+

L

⊗K+

R ≃ LRE/R (by [34, II.2.2.1]). Consequently, it suffices to show thatHi(LRE/R) = 0 for i ≥ 2.However,R = (K+/p1)⊗K+ R, andRE = (E+/q1)⊗K+ R; moreover,E+/q1 is the integralclosure of the valuation ringK+/p1 in the finite field extensionFrac(E+/q1) of Frac(K+/p1).Therefore we can replaceK+ by K+/p1 andE+ by K+/q1. This turns us back to the casewherer = 1. Then the vanishing ofHi(LE+/K+) for i ≥ 2 yields the vanishing ofHi(LRE/R)for i > 2. Moreover, sinceH1(LE+/K+) is torsion-free, multiplication bya onH1(LE+/K+) isinjective, thereforeH2(LRE/R) vanishes as well.

5.4. Logarithmic differentials. In this sectionK+ is a valuation ring of arbitrary rank. Wekeep the notation of (5.2.5.2.1). We start by reviewing somefacts on logarithmic structures, forwhich the general reference is [35].

5.4.5.4.1. LetMndX (reps.Z-ModX) be the category of sheaves of commutative monoids(resp. of abelian groups) on a topological spaceX. The forgetful functorZ-ModX →MndXadmits a left adjoint functorM 7→M gp. If M is a sheaf of monoids,M gp is the sheaf associatedto the presheaf defined by :U 7→ M(U)gp for every open subsetU ⊂ X.

The functorΓ : MndX →Mnd that associates to every sheaf of monoids its global sections,admits a left adjointMnd → MndX : M 7→ MX . For a monoidM , MX is the sheafassociated to the constant presheaf with valueM .


5.4.5.4.2. Recall that apre-log structureon a schemeX is a morphism of sheaves of commu-tative monoids :α : M → OX , where the monoid structure ofOX is induced by multiplicationof local sections. We denote bypre-logX the category of pre-log structures onX.

To a monoidM and a morphism of monoidsφ : M → Γ(X,OX), one can associate apre-log structureφX : MX → OX by composing the induced morphism of constant sheavesMX → Γ(X,OX)X with the counit of the adjunctionΓ(X,OX)X → OX .

5.4.5.4.3. To a morphismφ : Y → X of schemes, one can associate a pair of adjoint functorsφ∗ : pre-logX → pre-logY andφ∗ : pre-logY → pre-logX . Let (M,α : M → OX)(resp. (N, β : N → OY )) be a pre-log structure onX (resp. onY ) andφ : OX → φ∗OY

φ♯ : φ−1OX → OY the natural morphisms (unit and counit of the adjunction(φ−1, φ∗) on

sheaves ofZ-modules); thenφ−1Mφ−1α−→ φ−1OX

φ♯

−→ OY definesφ∗(M,α : M → OX) andφ∗(N, β : N → OY ) is the morphism of sheaves of monoidsγ : φ∗N ×φ∗OY

OX → OX whichmakes commute the cartesian diagram

φ∗N ×φ∗OYOX

γ //

OX

φ

φ∗N

φ∗β // φ∗OY .

5.4.5.4.4. A pre-log structureα is said to be alog structureif α−1(O×X) ≃ O×X . We denote by

logX the category of log structures onX. The forgetful functorlogX → pre-logX : M 7→Mpre-log admits a left adjoint

pre-logX → logX : M 7→M log(5.4.5)

and the resulting the diagram:

α−1(O×X)//

M

O×X

// M log

(5.4.6)

is cocartesian in the category of pre-log structures. From this, one can easily verify that the unitof the adjunction :M 7→ (Mpre-log)log is an isomorphism for every log structureM .

5.4.5.4.7. The categorylogX admits arbitrary colimits; indeed, since the unit of the adjunc-tion (5.4.5) is an isomorphism, it suffices to construct suchcolimits in the category of pre-logstructures, and then apply the functor(−) 7→ (−)log which preserves colimits, since it is a leftadjoint. In particular,logX admits arbitrary direct sums, and for any family(M i | i ∈ I) ofpre-log structures we have(⊕i∈IM i)

log ≃ ⊕i∈IMlogi .

5.4.5.4.8. For any morphism of schemesY → X we remark that, if(M,α) is a log structureonY , then the pre-log structureφ∗(M,α) is actually a log structure (this can be checked on thestalks). We deduce a pair of adjoint functors(φ∗, φ∗) for log structures, as in (5.4.5.4.3). Theseare formed by composing the corresponding functors for pre-log structures with the functor(5.4.5).

5.4.5.4.9. We say that a log structureM is regular if M = (MX)log for some free monoid

M , and the associated morphism of monoidsφ : M → Γ(X,OX) mapsM into the set ofnon-zero-divisors ofΓ(X,OX).


5.4.5.4.10. For anOX-moduleF , denote byHomOX(F , ∗) the category of all homomor-

phisms ofOX-modulesF → A (for any O-moduleA ). A morphism fromF → A toF → B is a morphismA → B of OX-modules which induces the identity onF . Thiscategory admits arbitrary colimits.

5.4.5.4.11. Given a pre-log structureα : M → OX , one defines the sheaf oflogarithmicdifferentialsΩX/Z(logM) as the quotient of theOX-moduleΩX/Z⊕ (OX⊗ZX

M gp) by theOX-submodule generated by the local sections of the form(dα(m),−α(m) ⊗ m), for every localsectionm of M . (The meaning of this is, that one adds toΩX/Z the logarithmic differentialsα(m)−1dα(m)). For every local sectionm of M , we denote byd log(m) the image of1⊗m inΩX/Z(logM). The assignmentM 7→ (ΩX/Z → ΩX/Z(logM)) defines a (covariant) functor :

Ω : pre-logX → HomOX(ΩX/Z, ∗).

Lemma 5.4.12.LetX be a scheme.

(i) The functorΩ commutes with all colimits.(ii) The functorΩ factors through the functor(5.4.5).

(iii) Let j : U → X be a formallyetale morphism of schemes andM a log structure onX.Then the natural morphism:j∗ΩX/Z(logM)→ ΩU/Z(log j

∗M) is an isomorphism.(iv) If M is a regular log structure, thenΩ(M) is a monomorphism ofOX-modules.

Proof. (i): It is clear thatΩ commutes with filtered colimits. Thus, to show that it commuteswith all colimits, it suffices to show that it commutes with finite direct sums and with coequal-izers. We consider first direct sums. We have to show that, forany two pre-log structuresM 1

andM 2, the natural morphism

ΩX/Z(logM 1) ∐ΩX/Z

ΩX/Z(logM 2)→ ΩX/Z(logM 1 ⊕M 2)

is an isomorphism. Notice that the functor(−) 7→ (−)gp of (5.4.5.4.1) commutes with colimits,since it is a left adjoint. It follows that the diagram

ΩX/Z //

ΩX/Z ⊕ (OX ⊗ZXM gp

1 )

ΩX/Z ⊕ (OX ⊗ZX

Mgp2 ) // ΩX/Z ⊕ (OX ⊗ZX

(M 1 ⊕M 2)gp)

is cocartesian. Thus, we are reduced to show that the kernel of the map

ΩX/Z ⊕ (OX ⊗ZXM gp)→ ΩX/Z(logM)

is generated by the images of the kernels of the corresponding maps relative toM 1 andM 2.However, any section ofM1⊕M 2 can be written locally in the formx · y for two local sectionsx of M1 andy of M2. Then we have :

(dα(x · y),−α(x · y)⊗ (x · y)) = (α(x) · dα(y) + α(y) · dα(x),−α(x) · α(y)⊗ (x · y))= α(x) · (dα(y),−α(y)⊗ y) + α(y) · (dα(x),−α(x)⊗ x)

so the claim is clear. Next, suppose thatφ : M → N are two morphisms of pre-log structures.Let : α : Q→ OX be the coequalizer ofφ andψ. Clearly, the coequalizer ofQ is the coequalizerof φ andψ in the category of sheaves of monoids. The functorM 7→ M gp preserves colimits,so we are reduced to consider the cokernel ofβ := φgp − ψgp. Moreover, clearly we haveCoker(β)⊗ZX

OX ≃ Coker(β ⊗ZXOX); the claim follows easily.

(ii): Let us apply the functorΩ to the cocartesian diagram (5.4.6). In view of (i), the resultingdiagram ofOX-modules is cocartesian. However, it is easy to check thatΩX/Z(logα

−1(O×X)) ≃ΩX/Z(logO

×X) ≃ ΩX/Z. The assertion follows directly.

(iii): one uses [21, Ch.IV, Cor. 17.2.4]; the details will beleft to the reader.


(iv): By (ii), the functorΩ descends to a functor

logX → HomOX(ΩX/Z, ∗).(5.4.13)

Since the unit of the adjunction (5.4.5) is an isomorphism, it follows easily that (5.4.13) com-mutes with all colimits of log structures. Hence, to verify thatΩ(M) is a monomorphism whenM is regular, we are immediately reduced to the case whenM is the regular log structure asso-ciated to a morphism of monoidsφ : N→ Γ(X,OX). Let f := φ(1). It is easy to check that inthis case, the diagram

OX

f //

df

OX

d log f

ΩX/Zβ // ΩX/Z(logM)

is cocartesian. By assumption,f is a non-zero-divisor, thus multiplication byf is a monomor-phism ofOX-modules, so the assertion follows.

5.4.5.4.14. This general formalism will be applied here to the following situation. We considerthe submonoidM := K+ \ 0 of K+. The imbeddingM ⊂ K+ induces a log structure onSpec(K+), which we call thetotal log structureonK+. More generally, we consider the naturalprojectionπ : M → Γ+ (see (5.1.5.1.25)); then for every submonoidN ⊂ Γ+, we have a logstructureN corresponding to the imbeddingπ−1(N) ⊂ K+. To ease notation, we will denotebyΩK+/Z(logN) the correspondingK+-module of logarithmic differentials.

Proposition 5.4.15. In the situation of(5.4.5.4.14), let ∆ ⊂ Γ be any subgroup,N a primeideal of∆+ (cp. (5.1.5.1.27)) and suppose that the convex rank ofΣ := ∆/(∆+ \N)gp equalsone. Then we have a short exact sequence

0→ ΩK+/Z(log∆+ \N)

j→ ΩK+/Z(log∆

+)ρ→ Σ⊗Z (K+/π−1(N) ·K+)→ 0.

Proof. Let us first remark that the assumptions and the notation makesense : indeed, sinceNis a prime ideal of∆+, it follows thatM := ∆+ \ N is a convex submonoid of∆+, henceM = (Mgp)+ andMgp is a convex subgroup of∆ (cp. (5.1.5.1.26)), thereforeΣ is an orderedgroup (cp. example (5.1.21)(v)), and hence it makes sense tosay that its convex rank equalsone.

Let us show thatj is injective. We can write∆+ as the colimit of the filtered family of itsfinitely generated submonoidsFα. For each suchFα, theorem 5.1.31 gives us a free finitelygenerated submonoidLα ⊂ ∆+ such thatFα ⊂ Lα. Clearly∆+ is the colimit of theLα, andMis the colimit of theMα :=M∩Lα. Thus∆+ is the colimit of theLα andM is the colimit of theMα. LetSα be a basis ofLα. SinceM is convex in∆, we see thatMα is free with basisSα∩MandLα =Mα⊕Nα, whereNα is the free submonoid spanned bySα \M . For eache ∈ Sα \M ,pick arbitrarily an elementxe ∈ K+ such that|xe| = e. The mape 7→ xe can be extendedto a map of monoidsNα → K+, and then to a pre-log structureνα : (Nα)SpecV → OSpecK+.Clearly we have an isomorphism of pre-log structures:Lα =Mα ⊕ να. Since the formation oflogarithmic differentials commutes with colimits of monoids, we are reduced to showing thatthe analogous map

jα : ΩK+/Z(logMα)→ ΩK+/Z(logLα)

is injective. By lemma 5.4.12(i), we haveKer(jα) ≃ Ker(ΩK+/Z → ΩK+/Z(log να)). Bylemma 5.4.12(iv), the latter map is injective, whence the assertion.

Next we proceed to show how to constructρ. Define a map

ρ : X := ΩK+/Z ⊕ (π−1∆⊗Z K+)→ Σ⊗Z (K+/π−1(N) ·K+)

by the rule :(ω, a⊗ b) 7→ π(a)⊗ b, for anyω ∈ ΩK+/Z, a ∈ π−1∆, b ∈ K+.


Claim 5.4.16. Ker(ρ) contains the kernel of the surjectionX → ΩK+/Z(log∆+).

Proof of the claim:It suffices to show thata ⊗ |a| ∈ Ker(ρ) wheneverπ(a) /∈ (∆+ \ N)gp.However,π(a) /∈ (∆+ \ N)gp ⇔ π(a) /∈ ∆+ \N ⇔ π(a) ∈ N ⇔ a ∈ π−1(N), so the claimfollows.

By claim 5.4.16 we deduce thatρ descends to the mapρ as desired. It is now obvious thatρ is surjective and that its kernel contains the image ofj. To conclude the proof, it sufficesto show that the cokernel ofj is annihilated byπ−1N . We are thus reduced to showing thatπ−1(N) annihilates the classes inCoker(j) of the elementsd log(e), for everye ∈ π−1N . Leta ∈ π−1(N). Since the convex rank ofΣ equals one, andπ(e) ∈ N , there existsk ≥ 0andb ∈ K+ such thate = ak · b and |b| < |a|. In particular,|b| ∈ ∆+, and we can write:d log(e) = d log(ak · b) = b · k · d log(a) + ak · d log(b), and it is clear thata annihilates each ofthe terms of this expression.

Corollary 5.4.17. In the situation of(5.4.5.4.14), we have :

(i) The natural map :βK+ : ΩK+/Z → ΩK+/Z(log Γ+) is injective.

(ii) Suppose moreover thatK+ has finite rank. Letpr := 0 ⊂ pr−1 ⊂ ... ⊂ p0 := mK bethe chain of all the prime ideals ofK+. Denote by∆r := ΓK ⊃ ∆r−1 ⊃ ... ⊃ ∆0 := 0the corresponding ascending chain of convex subgroups ofΓK (see(5.1.5.1.22)). ThenCoker(βK+) admits a finite filtrationFil•(Coker βK+) indexed by the totally ordered setSpec(K+), such that :

grpi(Coker βK+) ≃ (∆i+1/∆i)⊗Z (K+/pi) for everypi ∈ Spec(K+).

Proof. (i): Since the formation of differentials and logarithmic differentials commutes withcolimits ofZ-algebras and log structures, we can reduce to the case whereK is a field of finitetype over its prime field. In this case the convex rank ofΓ is finite, so the assertion follows fromproposition 5.4.15 and an easy induction.

(ii): is a straightforward consequence of proposition 5.4.15.

5.5. Transcendental extensions.In this section we extend the results of section 5.3 to the caseof arbitrary extensions of valued fields.

5.5.5.5.1. We fix the following notation throughout this section. For a valued field extension(K, | · |) ⊂ (E, | · |E), we let

ρE+/K+ : E+ ⊗K+ ΩK+/Z(log Γ+)→ ΩE+/Z(log Γ

+E)

be the natural morphism. One of the main results of this section states thatCoker(ρE+/K+) isinjective with torsion-free cokernel whenK is algebraically closed (theorem 5.5.20) or whenKhas characteristic zero (lemma 5.5.12). Lurking behind theresults of this sections there shouldbe some notion of ”logarithmic cotangent complex”, which however is not currently available.

5.5.5.5.2. LetG := (G, j,N,≤) be a datum as in (5.1.5.1.5). We wish to study the total logstructure of the valued field(K(G), | · |G). We consider the morphism of monoids

G→ K[G] : g 7→ [g].(5.5.3)

Let K[G]+ be the subring of the elementsx ∈ K[G] such that|x|G ≤ 1. Let π : G → ΓG

be the projection; for every submonoidM ⊂ Γ+G, the preimageπ−1M is a submonoid ofG,

and the restriction of (5.5.3) induces a morphism of monoidsπ−1(M) → K[G]+, whence apre-log structureπ−1MX on X := Spec(K[G]+) (see (5.4.5.4.2)). To ease notation, we setM := (π−1MX)

log and we will writeΩX/Z(logM) for the associated sheaf of log differentials.


Lemma 5.5.4. Resume the notation of(5.1.5.1.5). Then the natural diagram

K× ⊗Z K[G]+α //

β

G⊗Z K[G]+

η

ΩK+/Z(log Γ+)⊗K+ K[G]+ // ΩK[G]+/Z(log Γ

+G)

is cocartesian.

Proof. Let P be the push out ofα andβ. We already have a mapφ : P → ΩK[G]+/Z(log Γ+G),

and by inspecting the definition ofK[G]+ one verifies easily thatφ is surjective; thus we needonly find a left inverse forφ. Let us remark also thatβ, and consequentlyη, is surjective, henceit suffices to exhibit:(a) a derivationδ : K[G]+ → P such thatη(δa) = da for everya ∈ K[G]+;(b) aZ-linear mapψ : (π−1Γ+

G)gp → G such thatη ψ(γ) = d log(g) for everyg ∈ π−1Γ+

G.

Of course we can take forψ the natural identification(π−1Γ+G)

gp ∼→ G. To defineδ, choose

arbitrarily a set of representatives(gγ | γ ∈ G/K×) for the classes ofG/K×. Then everya ∈ K[G] can be written in a unique way as aK-linear combinationa =

∑γ∈G/K× aγ · [gγ];

we defineδ′ : K[G]+ → G⊗Z K[G]+ by the rule:a 7→∑

γ∈G/K× gγ ⊗ aγ . It is easy to checkthat the image ofδ′(a) in P does not depend on the choices of representatives, and this definesour sought derivationδ.

5.5.5.5.5. LetK(G)+ be the valuation ring of the valuation|·|G. It is easy to see thatK(G)+ =K[G]+p , wherep is the ideal of elementsx ∈ K[G] such that|x|G < 1. It then follows fromlemma 5.4.12(i),(iii) that the diagram of lemma 5.5.4 remains cocartesian when we replaceeverywhereK[G]+ byK(G)+.

Proposition 5.5.6. Suppose thatK is algebraically closed, let(E, | · |E) be a purely transcen-dental valued field extension of(K, | · |) with tr.deg(E : K) = 1. Then:

(i) ΩE+/K+ is a torsion-freeE+-module andHi(LE+/K+) = 0 for everyi > 0.(ii) The natural map ofE+-modules:ΩK+/Z(log Γ

+) ⊗K+ E+ → ΩE+/Z(log Γ+E) is injective

with torsion-free cokernel.(iii) Suppose thatΓE = Γ. Then the natural diagram

ΩK+/Z ⊗K+ E+ //

ΩE+/Z

ΩK+/Z(log Γ+)⊗K+ E+

ρE+/K+// ΩE+/Z(log Γ

+E)

is cocartesian.

Proof. Let mE be the maximal ideal ofE+ andX ∈ E such thatE = K(X). Following(5.1.5.1.10), we distinguish two cases, according to whether there exists or there does not existan elementa ∈ K which minimizes the functionK → ΓE : b 7→ |X − b|E . Suppose firstthat such an element does not exist. We pick a net(ai | i ∈ I) satisfying the conditions oflemma 5.1.9, relative to the elementx := X. For a givenb ∈ K, choosei ∈ I such that|X − ai|E < |X − b|E; then we have|ai − b| = |X − b|E and it follows easily thatΓE = ΓKin this case. Then, for everyi ∈ I we can findbi ∈ K such that|X − ai|E = |bi|. Letf(X)/g(X) ∈ E+ be the quotient of two elementsf(X), g(X) ∈ K[X ]. By lemma 5.1.9,we have|f(X)/g(X)|(ai,|bi|) ≤ 1 andγ := |g(X)|E = |g(X)|(ai,|bi|) for every sufficiently largei ∈ I. Picka ∈ K such that|a| = γ. Arguing as in the proof of case (b) of proposition 5.3.11(i),we deduce thata−1 · g(X), a−1 · f(X) ∈ Ai := K+[(X − ai)/bi], and if we letpi := Ai ∩mE+ ,thenf(X)/g(X) ∈ E+

i := Ai,pi. It is also easy to see that the family of theK+-algebrasE+i is


filtered by inclusion. ClearlyΩE+i /K

+ is a freeE+i -module of rank one, andHi(LE+

i /K+) = 0

for everyi > 0, so (i) follows easily in this case. Notice that, sinceΓ = ΓE , the log structureΓ+E on Spec(E+) (notation of (5.4.5.4.14)) is the log structure associatedto the morphism of

monoids(K+) \ 0 → E+. It follows easily that, for everyi ∈ I, we have a cocartesiandiagram

ΩK+/Z ⊗K+ Aiαi //

ΩAi/Z

ΩK+/Z(log Γ

+)⊗K+ Ai // ΩAi/Z(log Γ+)

(5.5.7)

where moreover,αi is split injective; the diagram of (iii) is obtained from (5.5.7), by localizingat p and taking colimits over the familyI; since both operations preserve colimits, we get (ii)and (iii) in this case. Finally, suppose that there exists anelementa ∈ K such that|X − a|is minimal; we can replaceX by X − a, and thus assume thata = 0. By (5.1.5.1.10) itfollows that | · |E is a Gauss valuation; then this case can be realized as the valuation | · |Gassociated to the datumG := (K× ⊕ Z, j, N,≤), wherej is the obvious imbedding, andNis eitherZ or 0, depending on whether|X|E ∈ Γ or otherwise. In either case, (5.5.5.5.5)tells us that the map of (ii) is split injective, with cokernel isomorphic toE+, so (ii) holds.Suppose first that|X|E ∈ Γ. Then we can findb ∈ K such that|X/b|E = 1, and one verifieseasily thatE+ is the localization ofA := K+[X/b] at the prime idealmK · E+. Clearly(5.5.7) remains cocartesian when we replaceAi byA andαi by the corresponding mapα; thelatter is still split injective, so (iii) follows easily. (i) is likewise obvious in this case. In case|X|E /∈ Γ, we distinguish three cases. First, suppose that|X|E < |b| for everyb ∈ K×. ThenK+[X/b] ⊂ E+ for everya ∈ K×, and indeed it is easy to check thatE+ is the filtered unionof itsK+-subalgebras of the formK+[X/b]pb , wherepb is the prime ideal generated bymK andX/b. Again (i) follows. The second case, when|X|E > |b| for everyb ∈ K×, is reduced tothe former, by replacingX with X−1. It remains only to consider the case where there exista0, b0 ∈ K such that|a0| < |X|E < |b0|; then we can find a net(ai, bi | i ∈ I) consistingof pairs of elements ofK×, such that|ai| < |X|E < |bi| for everyi ∈ I, and moreover, foreverya, b ∈ K× such that|a| < |X| < |b|, there existsi0 ∈ I with |a| < |ai| and |bi| < |b|wheneveri ≥ i0. In such a situation, one verifies easily thatE+ is the filtered union of itsK+-subalgebras of the formE+

i := K+[ai/X,X/bi]pi, wherepi is the prime ideal generatedby mK and the elementsai/X, X/bi. Each ringE+

i is a complete intersectionK+-algebra,isomorphic toK+[X, Y ]/(X ·Y −ai/bi). It follows thatLE+

i /K+ is acyclic in degrees> 0, and

ΩE+i /K

+ ≃ X ·E+i ⊕ Y ⊕E

+i /(XdY + Y dX). We leave to the reader the verification that this

E+i -module is torsion-free.

Theorem 5.5.8.Let (K, | · |) ⊂ (E, | · |E) be any extension of valued fields. Then(i) Hi(LE+/K+) = 0 for everyi > 1 andH1(LE+/K+) is a torsion-freeE+-module.(ii) If K is perfect, thenHi(LE+/K+) = 0 for everyi > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, we reduceeasily to the case whereE is finitely generated overK. Then, ifK is perfect, we can find a subextensionF ⊂ E whichis purely transcendental overK, and such thatE is separable overF ; by transitivity, we deducethatLE/K ≃ E⊗F LF/K ; moreoverHi(LF/K) = 0 for i > 0; by (i) we know thatH1(LE+/K+)imbeds intoH1(LE+/K+)⊗E+ E ≃ H1(LE/K), so the assertion follows.

To show (i), let| · |Ea be a valuation on the algebraic closureEa of E, which extends| · |E;recall thatEa+ is a faithfully flatK+-module by remark 5.1.12(ii). We apply transitivity to thetowerK+ ⊂ E+ ⊂ Ea+ to see that the theorem holds for the extension(K, | · |) ⊂ (E, | · |E)if and only if it holds for (K, | · |) ⊂ (Ea, | · |Ea) and for (E, | · |E) ⊂ (Ea, | · |Ea). For


the latter extension the assertion is already known by theorem 5.3.28(i), so we are reduced toprove the theorem for the case(K, | · |) ⊂ (Ea, | · |Ea). Similarly, we apply transitivity to thetowerK+ ⊂ Ka+ ⊂ Ea+ to reduce to the case where bothK andE are algebraically closed.Then we can writeE as the filtered union of the algebraic closuresEa

i of its finitely generatedsubfieldsEi, thereby reducing to prove the theorem for the extensionsK ⊂ Ea

i ; hence we canassume thattr.d(E : K) is finite. Again, by transitivity, we further reduce to the case wherethe transcendence degree ofE overK equals one. In this case, we can pick an elementX ∈ Etranscendent overK, and writeE = K(X)a. Using once more transitivity, we reduce to showthe assertion for the purely transcendental extensionK ⊂ K(X), in which case proposition5.5.6(i) applies, and concludes the proof.

Lemma 5.5.9. LetR→ S be a ring homomorphism.

(i) Suppose thatFp ⊂ R, denote byΦR : R → R the Frobenius endomorphism ofR, anddefine similarlyΦS. Let R(Φ) := Φ∗RR and S(Φ) := Φ∗SS (cp. (3.4.3.4.7)). Supposemoreover that the natural morphism :

R(Φ)

L

⊗R S → S(Φ)

is an isomorphism inD(R-Mod). ThenLS/R ≃ 0 in D(s.S-Mod).(ii) Suppose thatS is a flatR-algebra and letp be a prime integer,b ∈ R a non-zero-divisor

such thatp · R ⊂ bp · R. Suppose moreover that the Frobenius endomorphisms ofR′ :=R/bp · R andS ′ := S/bp · S are surjective. Then the natural morphism :

LS/R → LS[b−1]/R[b−1]

is an isomorphism inD(s.S-Mod).

Proof. (i): Let P • := P •R(S) be the standard simplicial resolution ofS by freeR-algebras. ThenLS/R ≃ ΩP •/R ⊗P • S. LetΦP • : P • → P •(Φ) be the termwise Frobenius endomorphism of thesimplicial algebraP •. As usual, we can writeΦP • = (ΦR ⊗R 1P •) ΦP •/R, where the relativeFrobeniusΦP •/R : R(Φ) ⊗R P • → P •(Φ) is a morphism of simplicialR(Φ)-algebras. Concretely,if P k = R[Xi | i ∈ I] is a free algebra on generators(Xi | i ∈ I), then

ΦP k/R(Xi) = Xpi for everyi ∈ I.(5.5.10)

Under the assumption of the lemma,ΦP •/R is a quasi-isomorphism of simplicialR(Φ)-algebras.It then follows from [34, Ch.II, Prop.1.2.6.2] thatΦP •/R induces an isomorphism

R(Φ) ⊗R LS/R∼→ LS(Φ)/R(Φ)

≃ LS/R.(5.5.11)

However, (5.5.10) shows that (5.5.11) is represented by a map of simplicial complexes which istermwise the zero map, so the claim follows.

(ii): Under the stated assumptions, the Frobenius map induces an isomorphism ofR-algebras:R/b · R

∼→ R′(Φ) (resp. ofS-algebras:S/b · S

∼→ S ′(Φ)). Thus the map

S ′ ⊗R′ (R′)(Φ) → S ′(Φ) : x⊗ y 7→ ΦS′(x) · y

is an isomorphism. Since moreoverS ′ is a flatR′-algebra, we see that the assumption of (i) issatisfied, whenceLS′/R′ ≃ 0. If we now tensor the short exact sequence0→ R→ R→ R′ →0 by LS/R, we obtain a distinguished triangle

LS/Rbp−→ LS/R → LS/R

L

⊗R R′ → σLS/R.

However,LS/RL

⊗R R′ ≃ LS′/R′ by [34, II.2.2.1], so we have shown thatbp acts as an isomor-phism onLS/R. In other words,LS/R ≃ LS/R ⊗S S[b−1] ≃ LS[b−1]/R[b−1], as claimed.


Lemma 5.5.12.Let (K, | · |) ⊂ (E, | · |E) be an extension of valued fields and suppose thatQ ⊂ κ(K). Then the mapρE+/K+ of (5.5.5.5.1)is injective with torsion-free cokernel.

Proof. To begin with, let(F, | · |F ) be any valued field extension of(E, | · |E). We remark that:

ρF+/E+ (ρE+/K+ ⊗E+ 1F+) = ρF+/K+.(5.5.13)

Claim 5.5.14. Suppose moreover thatE is an algebraic extension ofK. ThenρE+/K+ is anisomorphism.

Proof of the claim:Applying (5.5.13), withF := Ea, we reduce easily to prove the claim incaseE is algebraically closed. LetKsh be the field of fractions of the strict henselization ofK+

(which we see as imbedded inE+). Let j : Spec(Ksh+)→ Spec(K+) be the morphism inducesby the imbeddingK ⊂ Ksh; In view of lemma 5.2.5, the log structureΓ+

Ksh on Spec(Ksh+)

(notation of (5.4.5.4.14)) equalsj∗Γ. Since moreoverKsh+ is local ind-etale overK+, wededuce from 5.4.12(iii) thatρKsh+/K+ is an isomorphism. Then arguing as in the foregoing,we see that it suffices to prove the claim for the case whenK = Ksh. Since everything insight commutes with filtered unions of field extensions, we can even reduce to the case whereE is a finite (Galois) extension ofK. Then, by corollary 5.2.14, this case can be realizedas the extension associated to some datumG := (G, j,N,≤), where moreoverG/K× is afinite torsion group. Since by assumptionQ ⊂ K[G]+, the claim follows by lemma 5.5.4 and(5.5.5.5.5).

Now, if K ⊂ E is an arbitrary extension, we can apply (5.5.13) withF := Ea and claim5.5.14 to the extensionE ⊂ Ea to reduce to the case whereE is algebraically closed. Thenwe can apply again claim 5.5.14 to the extensionK ⊂ Ka and (5.5.13) withE := Ka andF := E, to reduce to the case where alsoK is algebraically closed. Then, by the usual argumentwe reduce to the case of an extension of finite transcendence degree, and even to the case oftranscendence degree equal to one. We factor the latter as a tower of extensionsK ⊂ K(X) ⊂E, whereX is transcendental overK, henceE algebraic overK(X). So finally we are reducedto the caseE = K(X), in which case we conclude by proposition 5.5.6(ii).

Theorem 5.5.15.Let(K, |·|) ⊂ (E, |·|E) be an extension of valued fields, withK algebraicallyclosed. ThenΩE+/K+ is a torsion-freeE+-module.

Proof. Pick a valuation| · |Ea of the fieldEa extending| · |E. We have an exact sequence:H1(LEa+/E+) → Ea+ ⊗E+ ΩE+/K+ → ΩEa+/K+, where the leftmost term is torsion-free bytheorem 5.3.28, so it suffices to show thatΩEa+/K+ is torsion-free, and we can therefore assumethatE is algebraically closed. In this case, if nowchar(K) > 0, it follows that the Frobeniusendomorphism ofE+ is surjective; then, for anya ∈ E+ we can writeda = d(a1/p)p =p · a(p−1)/p · da1/p = 0, so actuallyΩE+/K+ = 0. In casechar(K) = 0 andchar(κ(K)) =p, let us pick an elementb ∈ K+ such that|bp| ≥ |p|. SinceK andE are algebraicallyclosed, the Frobenius endomorphisms onK+/bpK+ andE+/bpE+ are surjective, soLE+/K+ ≃LE+[b−1]/K+[b−1] by lemma 5.5.9(ii). Now,K+[b−1] is the valuation ring of a valuation| · |′ onK, which extends to a valuation| · |′E onE+ whose valuation ring isE+[b−1]. Furthermore, theresidue fields of these valuations are fields of characteristic zero. Hence, we have reduced theproof of the theorem to the case whereκ(K) ⊃ Q. We can further reduce to the case wheretr.d(E : K) is finite andK is the algebraic closure of an extension of finite type of its primefield. By lemma 5.5.12 we have a commutative diagram with exact rows:

E+ ⊗K+ ΩK+/Z//

1E+⊗βK+

ΩE+/Z//

βE+

ΩE+/K+ //

γ

0

0 // E+ ⊗K+ ΩK+/Z(log Γ+) // ΩE+/Z(log Γ

+E)

// Coker(ρE+/K+) // 0


whereβK+ andβE+ are the maps of corollary 5.4.17(i). By virtue of lemma 5.5.12, it suffices toshow thatγ is injective. SinceβE+ is injective by corollary 5.4.17(i), the snake lemma reducesus to prove :

Claim5.5.16. The induced mapE+ ⊗K+ Coker βK+ → Coker βE+ is injective.

Proof of the claim:Under our current assumptions,K+ andE+ are valuation rings of finiteKrull dimension, by (5.1.5.1.24). Letpr := 0 ⊂ pr−1 ⊂ ... ⊂ p0 := mE be the chain of allthe prime ideals ofE+. Denote by∆r := ΓE ⊃ ∆r−1 ⊃ ... ⊃ ∆0 := 0 the correspond-ing ascending chain of convex subgroups ofΓE (see (5.1.5.1.22)). LetFil•(Coker βE+) (resp.Fil•(Coker βK+)) be the finite filtration indexed by the totally ordered setSpec(E+) (resp.Spec(K+)), provided by corollary 5.4.17(ii). Since it is preferableto work with a single index-ing set, we use the surjectionSpec(E+) → Spec(K+), to replace bySpec(E+) the indexingof the filtration onCoker βK+; of course in this way some of the graded subquotients becometrivial, but we do not mind. With this notation we can write down the identities:

E+ ⊗K+ grpi(Coker βK+) ≃ E+ ⊗K+ ((∆i+1 ∩ Γ)/(∆i ∩ Γ))⊗Z (K+/pi ∩K+)

for everypi ∈ Spec(E+). Furthermore, our mapφ : E+⊗K+Coker βK+ → Coker βE+ respectsthese filtrations. If nowpi ∈ Spec(E+) is the radical of the extension of a prime ideal ofK+,then clearly the mapgrpi(φ) : E+ ⊗K+ grpi(Coker βK+)→ grpi(Coker βE+) is induced by theimbeddings(∆i+1 ∩ Γ)/(∆i ∩ Γ) ⊂ ∆i+1/∆i andK+/pi ∩K+ ⊂ E+/pi, and it is thereforeinjective. On the other hand, ifpi is not the radical of an ideal extended fromK+, we havegrpi(Coker βK+) = 0, sogrpi(φ) is trivially injective in this case as well. Since the mapgr•(φ)is injective, the same holds forφ, which concludes the proof of the claim and of the theorem.

Theorem 5.5.17.Let | · |Ks be a valuation on the separable closureKs of K, extending thevaluation ofK. Then the mapρ := ρKs+/K+ is injective.

Proof. Suppose first thatΓ is divisible. By the usual reductions, we can assume thatK isfinitely generated over its prime field, hence that the convexrank ofΓKs is finite. We considerthe commutative diagram :

ΩK+/Z ⊗K+ Ks+ α //

βK+⊗1Ks+

ΩKs+/Z

βKs+

ΩK+/Z(log Γ+)⊗K+ Ks+ ρ // ΩKs+/Z(log Γ

+Ks)

whereβK+ andβKs+ are the maps of corollary 5.4.17(i). By theorem 5.3.28(ii),α is injective,and the same holds forβKs+, by corollary 5.4.17(i). It follows thatIm(βK+⊗1Ks+)∩Ker(ρ) =0, in other words, the induced map

Ker(ρ)→ Ks+ ⊗K+ Coker(βK+)(5.5.18)

is injective. By corollary 5.4.17(ii), there is a filtrationFil•(Coker βK+) on Coker(βK+), in-dexed bySpec(K+), such thatgrpi(Coker βK+) ≃ (∆i+1/∆i)⊗Z (K

+/pi), where∆i,∆i+1 aretwo convex subgroups ofΓK . However, since we assume thatΓ is divisible, the same holds for∆i+1/∆i; we deduce thatgrpi(Coker βK+) vanishes wheneverFrac(K+/pi) is a field of posi-tive characteristic. What this means is that the filtrationFil•(Coker βK+) is actually indexed bySpec(K+ ⊗Z Q) ⊂ Spec(K+), and the natural map :

Coker(βK+)→ Coker(βK+)⊗Z Q(5.5.19)

is an isomorphism. The same holds also forCoker(βKs+). If K+Q := K+ ⊗Z Q = 0, then

Coker(βK+) = Coker(βKs+) = 0, henceKer(ρ) = 0, which is what we had to show.


In caseK+Q 6= 0, thenK+

Q is a valuation ring ofK with residue field of characteristiczero. However, by lemma 5.4.12(iii) it follows easily thatCoker(βK+) ⊗Z Q ≃ Coker(βK+

Q),

and likewiseρKs+/K+ ⊗Z 1Q = ρKs+Q/K+

Q, whereKs+

Q := Ks+ ⊗Z Q is a valuation ring ofKs

whose valuation extends that ofK+Q . Since (5.5.19) is an isomorphism, (5.5.18) factors through

Ker(ρKs+Q/K+

Q); however, the latter vanishes by lemma 5.5.12. Since (5.5.18) is injective, we

deriveKer(ρ) = 0, so the theorem holds in this case.In caseΓ is not necessarily divisible, let us choose a datumG := (G, j,N,≤) as in (5.1.5.1.5),

such thatG := (Ks)× ⊕ F , whereF is a torsion-free abelian group (whose composition lawwe write in multiplicative notation) andN is the graph of a surjective group homomorphismφ : F → (Ks)×. Notice that in this caseΓG ≃ ΓKs ≃ Γ ⊗Z Q, and the restriction toF ofthe projectionG → ΓG is the mapx 7→ |φ(x−1)|Ks. Let nowH := K× ⊕ F and define anew datumH := (H, j,H ∩ N,≤); sinceφ is surjective, clearly we still haveΓH ≃ Γ ⊗Z Q.Notice as well thatKs(G) is separable overK(H). SetρH := ρK(H)+/K+, ρG := ρKs(G)+/Ks+

andρG/H := ρKs(G)+/K(H)+ . We consider the diagram :

ΩK+/Z(log Γ+)⊗K+ Ks(G)+

ρH⊗1Ks(G)+

ρ⊗1Ks(G)+ // ΩKs+/Z(log Γ+Ks)⊗Ks+ Ks(G)+

ρG

ΩK(H)+/Z(log Γ+H)⊗K(H)+ K

s(G)+ρG/H // ΩKs(G)+/Z(log Γ

+G).

SinceF is torsion-free, it follows easily from (5.5.5.5.5) and lemma 5.5.4 thatρH andρG areinjective with torsion-free cokernels. Hence, in order to prove thatρ is injective, it sufficesto show thatρKs(G)+/K(H)+ is. Finally, letE be the separable closure ofK(H) and choose avaluation onE which extends the valuation ofKs(G); we notice thatKer(Ks(G)+/K(H)+) ⊂Ker(ρE+/K(H)+). Therefore, we can replaceK by K(H) and reduce to the case whereΓ isdivisible, which has already been dealt with.

Theorem 5.5.20.Let (K, | · |) ⊂ (E, | ⊂ |E) be an extension of valued fields, withK alge-braically closed. ThenρE+/K+ is injective with torsion-free cokernel.

Proof. By the usual arguments, we can suppose thatE is finitely generated overK. Let | · |Es bean extension of the valuation| · |E to the separable closureEs of E. We haveKer(ρE+/K+) ⊂Ker(ρEs+/K+), and evenCoker(ρE+/K+) ⊂ Coker(ρEs+/K+), by theorem 5.5.17. Thus wecan replaceE by Es and suppose thatE is separably closed, henceΓE divisible, by example5.1.21(ii). By corollary 5.4.17(i) we have a commutative diagram with exact rows :

0 // E+ ⊗K+ ΩK+/Z

α

// E+ ⊗K+ ΩK+/Z(log Γ+) //

ρE+/K+

E+ ⊗K+ Coker(βK+) //

γ

0

0 // ΩE+/Z// ΩE+/Z(log Γ

+E)

// Coker(βE+) // 0.

By theorem 5.5.8(ii), the mapα is injective. The same holds forγ, in view of claim 5.5.16. Itfollows already thatρE+/K+ is injective. Moreover, by theorem 5.5.15,Coker(α) is a torsion-freeE+-module. Since bothΓ andΓE are divisible, it follows easily from corollary 5.4.17(ii)thatCoker(βK) andCoker(βE) areQ-vector spaces (cp. the proof of theorem 5.5.17), hencethe same holds forCoker(γ). Consequently,Coker(ρE+/K+) is a torsion-freeZ-module, andthus we are reduced to show thatQ⊗Z Coker(ρE+/K+) is a torsion-freeE+-module. However,Q⊗ZCoker(ρE+/K+) ≃ Coker(ρE+

Q/K+

Q), whereE+

Q := E+⊗ZQ andK+Q := K+

Q are valuationrings with residue fields of characteristic zero (or else they vanish, in which case we are done).But the assertion to prove is already known in this case, by lemma 5.5.12.


Corollary 5.5.21. Let (K, | · |) be a valued field, andk a perfect field such thatk ⊂ K+. ThenΩK+/k andCoker(ρK+/k) are torsion-freeK+-modules.

Proof. We haveka ⊂ Ksh+; letE := ka ·K ⊂ Ksh and denote byj : Spec(E+)→ Spec(K+)the morphism induced by the imbeddingK ⊂ E. By lemma 5.2.5 the natural map :j∗Γ+ → Γ+

E

is an isomorphism of log structures; moreoverΩka/k = 0, sincek is perfect. HenceΩK+/k ⊂ΩE+/ka , and furthermore, by lemma 5.4.12(iii) we haveCoker(ρK+/k) ⊂ Coker(ρE+/ka). Thenthe assertion follows from theorems 5.5.15 and 5.5.20.

Remark 5.5.22. Notice that corollary 5.5.21 is a straightforward consequence of a standard (asyet unproven) conjecture on the existence of resolution of singularities over perfect fields.

5.6. Deeply ramified extensions.We keep the notation of section 5.3. We borrow the notionof deeply ramified extension of valuation rings from the paper [14], even though our definitionapplies more generally to valuations of arbitrary rank.

Definition 5.6.1. Let (K, | · |) be a valued field,| · |Ks a valuation onKs which extends| · |. Wesay that(K, | · |) is deeply ramifiedif ΩKs+/K+ = 0. Notice that the definition does not dependon the choice of the extension| · |Ks.

Proposition 5.6.2. Let (K, | · |) be a valued field whose valuation has rank one,| · |Ks anextension of| · | toKs. Then the following conditions are equivalent :

(i) (K, | · |) is deeply ramified;(ii) The morphism of almost algebras(K+)a → (Ks+)a is weaklyetale;(iii) (ΩKs+/K+)a = 0.

Moreover, the above equivalent conditions imply that the valuation ofK is not discrete.

Proof. We leave to the reader the verification that (iii) (and,a fortiori, (i)) can hold only in casethe valuation ofK is not discrete.

Let K+sh be a strict henselization ofK+ contained inKs+, andKsh its fraction field. It iseasy to check that(K, | · |) is deeply ramified if and only if(Ksh, | · |Ksh) is. Moreover, in viewof lemma 3.1.2(iv), condition (ii) holds for(K, | · |) if and only if it holds for(Ksh, | · |Ksh),and similarly for condition (iii). Hence we can assume thatK+ is strictly henselian. It is alsoclear that (i)⇒(iii)⇐(ii). To show that (iii)⇒(ii), let E ⊂ Ks be a finite separable extension ofK and setE+ := Ks+ ∩ E; it follows from theorem 5.3.20 that the natural mapΩE+/K+ ⊗E+

Ks+ → ΩKs+/K+ is injective; thus, if (iii) holds, we deduce that(ΩE+/K+)a = 0 for everyfinite separable extensionE of K. Again by theorem 5.3.20 we derive thatDE+/K+ = E+a forevery suchE. Finally, lemmata 5.3.10(i) and 4.1.29 show thatE+a is etale overK+a, whence(ii). Suppose next that the residue characteristic ofK+ is zero; then every finite extensionEof K factors as a tower of Kummer extensions of prime degree, thereforeΩE+/K+ = 0 bycorollary 5.3.18(i), which implies that (i)⇔(iii) in this case. Finally, suppose that the residuecharacteristic isp > 0. Let us choosea ∈ K+ such that|a| ≥ |p|. It follows easily fromexample 5.1.21(iii) that every elementx ∈ Ks+ can be written in the formx = yp + a · z forsomey, z ∈ Ks+. Hencedx = p · dyp−1 + a · dz, which means thatΩKs+/K+ = a · ΩKs+/K+.Therefore, even in this case we deduce (iii)⇒(i).

5.6.5.6.3. Let us say that aK+a-moduleM is K+a-divisible if, for everyx ∈ K+ \ 0 wehaveM = x ·M .

Lemma 5.6.4. Let (K, | · |) be a valued field such thatQ ⊂ K, and letp := char(κ) > 0. Let(Ks, | · |Ks) be an extension of the valuation| · | to a separable closure ofK. Denote byT theKs+-torsion submodule ofΩKs+/Z. ThenT ≃ Ks/Ks+.


Proof. Let (Qa, | · |Qa) be the restriction of| · |Ks to the algebraic closure ofQ in Ks. Fromtheorem 5.5.20 it follows easily thatT ≃ ΩQa+/Z⊗Qa+Ks+, hence we can suppose thatK = Q.From theorem 5.3.20 we deduce that the natural mapQa+ ⊗E+ ΩE+/Z → ΩQa+/Z is injectivefor every subextensionE ⊂ Qa. For everyn ∈ N and every subextensionE ⊂ Qa, letEn := E(ζpn), whereζpn is any primitivepn-th root of1 and setE∞ :=

⋃n>0En.

Claim 5.6.5. For every finite subextensionE ⊂ Qa, there existsn ∈ N such that the image ofE+n ⊗E+ ΩE+/Z in ΩE+

n /Zis included in the image ofE+

n ⊗Q+nΩQ

+n /Z

.

Proof of the claim:For everyn ∈ N, E+n is a discrete valuation ring andκ(En) is a finite sepa-

rable extension ofκ(Q) = Fp; from the exact sequencemEn/m2En→ ΩE+

n /Z→ Ωκ(En)/Fp = 0

we deduce thatΩE+n /Z

is a (torsion) cyclicE+n -module. By comparing the annihilators of the

modules under consideration, one obtains easily the claim.

A standard calculation shows thatΩQ+∞/Z ≃ Q∞/Q+

∞. This, together with claim 5.6.5 impliesthe lemma.

Proposition 5.6.6. Keep the notation and assumptions of proposition 5.6.2 and suppose more-over that the characteristicp of the residue fieldκ ofK+ is positive and that the valuation onK is not discrete. Let(K∧, | · |∧) be the completion of(K, | · |) for the valuation topology. Thenthe following conditions are equivalent:

(i) (K, | · |) is deeply ramified;(ii) The Frobenius endomorphism ofK∧+/p ·K∧+ is surjective;

(iii) For someb ∈ K+ \ 0 such that1 > |b| ≥ |p|, the Frobenius endomorphism on(K+/b ·K+)a is an epimorphism;

(iv) ΩK+/Z(log Γ+) is aK+-divisibleK+-module;

(v) ΩK+/Z is aK+-divisibleK+-module;(vi) (ΩK+/Z)

a is aK+a-divisibleK+a-module;(vii) Coker(ρKs+/K+) = 0 (notation of (5.5.5.5.1));(viii) Coker(ρKs+/K+)a = 0.

Proof. Suppose that (i) holds; then by proposition 5.6.2 it followsthat the morphism(K+)a →(Ks+)a is weakly etale, so the same holds for the morphism(K+/b ·K+)a → (Ks+/b ·Ks+)a,for everyb ∈ K+ with |b| ≥ |p|. In view of example 5.1.21(iii), one sees that the Frobeniusendomorphism onKs+/b · Ks+ is an epimorphism. Using theorem 3.4.13(ii) we deduce thatthe Frobenius endomorphism on(K+/b · K+)a is an epimorphism as well. This shows that(i)⇒(iii). To show that (iii)⇒(ii), let us chooseε ∈ m \ 0 such that|εp| > |b|; by hypothesis,for everyx ∈ K+ there existsy ∈ K+ such thatεp · x − yp ∈ b · K+. It follows easilythat the Frobenius endomorphism is surjective onK+/(b · ε−p)K+. Replacingb by b/ε−p

we can assume that the Frobenius endomorphism is surjectiveon K+/bK+. Let b1 ∈ K+

such that1 > |bp1| ≥ |b|; we letFil•1(K+/pK+) (resp. Fil•2(K

+/pK+)) be theb1-adic (resp.bp1-adic) filtration onK+/pK+. The group topology onK+/pK+ defined by the filtrationsFil•i (K

+/pK+) (i = 1, 2) is the same as the one induced by the valuation topology ofK+;moreover, one verifies easily that the Frobenius endomorphism defines a morphism of filteredabelian groupsFil•1(K

+/pK+)→ Fil•2(K+/pK+) and that the associated morphism of graded

abelian groups is surjective. It then follows from [11, Ch.III, §2, n.8, Cor.2] that (ii) holds.Next suppose that (ii) holds; chooseb ∈ K+ such that1 > |b| > |bp| ≥ |p|; by hypothesis,the Frobenius endomorphism onK+/bpK+ is surjective; the same holds for the Frobenius mapon Ks+/bpKs+, in view of example 5.1.21(iii). Hence, the assumptions of lemma 5.5.9(ii)are fulfilled, and we deduce thatLKs+/K+ ≃ LKs+[b−1]/K+[b−1]. Now, if char(K) = p, thisimplies already thatΩKs+/K+ ≃ ΩKs/K = 0, which is (i). In casechar(K) = 0, we onlydeduce thatΩKs+/K+ ≃ ΩKs+[1/p]/K+[1/p]; however,R := K+[1/p] is a valuation ring of residue


characteristic0. We are therefore reduced to showing thatR is deeply ramified. Arguing as inthe proof of proposition 5.6.2 we can even assume thatR is strictly henselian, in which case theassertion follows from corollary 5.3.18(i).

Furthermore, (ii) implies easily thatΩ(K+/bpK+)/Z = 0 (sincedxp = p · dxp−1 = 0). LetI := bp ·K+; it follows that the natural mapI/I2 → (K+/bpK+)⊗Z ΩK+/Z is surjective,i.e.ΩK+/Z = bp ·ΩK+/Z +K+ · dbp ⊂ bp ·ΩK+/Z, which implies (v). Next, by corollary 5.4.17(ii),we haveΩK+/Z(log Γ

+)/ΩK+/Z ≃ κ ⊗Z Γ, and this last term vanishes since (ii) implies thatp · Γ = Γp. This shows that (ii)⇒(iv) as well. Clearly (v)⇒(vi). Suppose that (vi) holds. Wewill need the following :

Claim5.6.7. ΩKs+/Z is aKs+-divisible module andC := Coker(ρKs+/K+) is aK+-torsionmodule (notation of (5.5.5.5.1)). Furthermore,Ca ≃ (ΩKs+/K+)a.

Proof of the claim:In view of example 5.1.21(iii),(Ks, | · |Ks) satisfies condition (iii), hence thefirst assertion follows from the implications (iii)⇔(ii)⇒(v), which have already been shown.Furthermore, it is clear thatΩKs+/K+ is a torsionK+-module and therefore the second asser-tion follows easily from corollary 5.4.17. The latter corollary also implies that(ΩK+/Z)

a ≃ΩK+/Z(log Γ

+)a, and similarly forΩKs+/Z, whence the third assertion.Now, suppose first thatFp ⊂ K+; in this caseΩKs+/Z is a torsion-freeKs+-module according

to corollary 5.5.21. Letb ∈ K+ be any element; by theorem 5.5.17 and snake lemma we deducethat theb-torsion submoduleC[b]a := Ker(Ca → Ca : x 7→ b · x) is isomorphic to the cokernelof the scalar multiplication byb on the moduleKs+ ⊗K+ ΩK+/Z(log Γ

+); the latter vanishesby assumption, and by claim 5.6.7 we haveCa =

⋃b∈K+ C[b]a, whenceCa = 0, which is

equivalent to (i) by claim 5.6.7 and proposition 5.6.2.Finally, in caseK+ is of mixed characteristic, denote byT (resp. T ′) theK+-torsion sub-

module ofΩKs+/Z (resp. ofKs+ ⊗K+ ΩK+/Z) and defineT [b] (resp. T ′[b]) as itsb-torsionsubmodule, for anyb ∈ K+. The foregoing argument shows thatT a is isomorphic to theK+a-torsion submodule of(ΩKs+/Z(log Γ

+Ks))a, and similarly for(T ′)a; moreover, by snake lemma

we obtain a short exact sequence0→ T ′[b]a → T [b]a → C[b]a → 0 for everyb ∈ K+, whencea short exact sequence0 → (T ′)a → T a → Ca → 0. Under (vi),(T ′)a is a divisible module;however, it is clear from lemma 5.6.4 that the only divisible(Ks+)a-submodules ofT a are0 andT a. Consequently, in light of claim 5.6.7 and proposition 5.6.2, in order to prove that (vi)⇒(i),it suffices to show that(T ′)a 6= 0. In turns, this is implied by the following :

Claim5.6.8. The image inΩK+/Z(log Γ) of d log(p) ∈ ΩQ+/Z(log Γ+Q), has annihilatorp ·K+.

Proof of the claim:(Of course,Q+ := Q+∩ Q). By theorem 5.5.17 it suffices to consider the

image ofd log(p) in ΩKs+/Z(log Γ+Ks). Then, by theorem 5.5.20, we reduce to consider the case

Ks = Q. Then, once more by theorem 5.5.17, it suffices to look at the annihilator ofd log(p) inΩQ+/Z(log Γ

+Q) itself, and the claim follows.

Since (vi) is implied by both (iv) and (v), we deduce at once that all the conditions (i)-(vi) areequivalent. Furthermore, it is clear from claim 5.6.7 and proposition 5.6.2 that both (vii) and(viii) are equivalent to (i), so the proposition follows.

Remark 5.6.9. By inspecting the proof of proposition 5.6.6, we see that theargument for(ii)⇒(i) still goes through for valued fields(K, | · |) of arbitrary rank and characteristicp > 0.

Lemma 5.6.10.Let (K, | · |) be a valued field andb ∈ K an element with0 < |b| < 1. Denoteby q(b) the radical of the idealb · K+ and setp :=

⋂r>0 b

r · K+. Thenp(b) and q(b) areconsecutiveprime ideals,i.e. there are no prime ideals strictly contained betweenp(b) andq(b). Equivalently, the ringW (b) := (K+/p(b))q(b) is a valuation ring of rank one and theimage ofb is topologically nilpotent in the valuation topology ofW (b).


Proof. It is easy to verify thatp(b) andq(b) are prime ideals, and using (5.1.5.1.22) one deducesthatW (b) is a valuation ring of rank one, which means thatp(b) andq(b) are consecutive,

Theorem 5.6.11.Let (K, | · |) be a valued field,(K∧, | · |∧) its completion. The followingconditions are equivalent :

(i) (K, | · |) is deeply ramified.(ii) For every valued extension(E, | · |E) of (K, | · |), for everyb ∈ K+ \ 0 and for every

i > 0 we haveHi(L(E+/bE+)/(K+/bK+)) = 0.(iii) For every pair of consecutive prime idealsp ⊂ q ⊂ K+, the valuation ring(K+/p)q is

deeply ramified.(iv) For every pair of convex subgroupsH1 ⊂ H2 ⊂ Γ, the quotientH2/H1 is not isomorphic

to Z, and moreover, ifp := char(κ) > 0, the Frobenius endomorphism onK∧+/p ·K∧+

is surjective.

Proof. To show that (ii)⇒(i), we takeE := Ks and we choose a valuation onKs extending| · |. Then, by arguing as in the proof of lemma 5.5.9(ii), we deduce from (ii) that the scalarmultiplication byb onΩKs+/K+ is injective. Since the latter is a torsionKs+-module, we deduce(i). To show (i)⇒(ii), we reduce first to the case whereE = Ea; indeed, let| · |Ea be avaluation onEa extending|·|E and suppose that the sought vanishing is known for the extension(K, | · |) ⊂ (Ea, | · |Ea); by transitivity, it then suffices to show :

Claim 5.6.12. Hi(L(Ea+/bEa+)/(E+/bE+)) = 0 for everyi > 1.

Proof of the claim:By [34, II.2.2.1] we haveL(Ea+/bEa+)/(E+/bE+) ≃ LEa+/E+

L

⊗Ea+Ea+/bEa+,whence a spectral sequence

E2pq := TorE

a+

p (Hq(LEa+/E+), Ea+/bEa+)⇒ Hp+q(L(Ea+/bEa+)/(E+/bE+)).

SinceEa+/bEa+ is anEa+-module of Tor-dimension≤ 1, we see thatE2pq = 0 for every

p > 1; furthermore, by theorem 5.3.28(i), it follows thatE2pq = 0 wheneverp, q > 0, so the

claim follows.Thus, we can suppose thatE is algebraically closed. A spectral sequence analogous to the

foregoing computesHi(L(E+/bE+)/(Ka+/bKa+)), and using theorems 5.5.8(ii) and 5.5.15, wefind that the latter vanishes fori > 0. Consequently, by applying transitivity to the towerof extensionsK+/bK+ ⊂ Ka+/bKa+ ⊂ E+/bE+, we reduce to show the assertion for thecaseE = Ka. However, by example 5.1.21(iii) we haveKa+/bKa+ ≃ Ks+/bKs+ for everyb ∈ K+ \ 0, so we can further reduce to the caseE = Ks. In this case, one concludesthe proof by another spectral sequence argument, this time using assumption (i) and theorem5.3.28(ii) to show that the relevant termsE2

pq vanish.To show that (iii)⇒(iv), we consider two subgroupsH1 ⊂ H2 as in (iv); if c.rk(H2/H1) > 1,

then clearlyH2/H1 cannot be isomorphic toZ, so we can assume thatH1 andH2 are consec-utive, so that the corresponding prime ideals are too (see (5.1.5.1.22)). In this case, (iii) andproposition 5.6.2 show thatH2/H1 is not isomorphic toZ, which is the first assertion of (iv).To prove the second assertion, it will suffice to show the following :

Claim 5.6.13. Suppose thatp := char(κ) > 0 and that (iii) holds. Then, for everyb ∈ K+\0with 1 > |b| ≥ |p|, the Frobenius endomorphism onK+/b ·K+ is surjective.

Proof of the claim:For such ab as above, definep(b), q(b) andW (b) as in lemma 5.6.10; thenW (b) is a valuation ring of rank one, so it is deeply ramified by assumption (iii). Then, byproposition 5.6.6 it follows that the Frobenius endomorphism is surjective onW (b)/b ·W (b) ≃K+

q(b)/b·K+q(b). We remark thatb·K+

q(b) ⊂ K+; there follows a natural imbedding:K+/b·K+q(b) ⊂

W (b)/b·W (b), commuting with the Frobenius maps. It is then easy to deducethat the Frobenius


endomorphism is surjective onK+/b · K+q(b). Moreover, by proposition 5.6.2, the valuation of

W (b) is not discrete, hence its value group is isomorphic to a dense subgroup of(R,≥) (seeexample 5.1.21(vi)); therefore, by (5.1.5.1.22) and example 5.1.21(v), we deduce that thereexists an elementc ∈ K+ such that|b| > |c3p| and |b| < |c2p|. These inequalities have beenchosen so thatcp·K+

q(b) ⊂ K+ andb·K+q(b) ⊂ c2p·K+

q(b), whenceb·K+q(b) ⊂ cp·K+, and finally we

conclude that the Frobenius endomorphism induces a surjection: K+/c ·K+ → K+/cp ·K+.We letFil•1(K

+/b · K+) (resp. Fil•2(K+/b · K+)) be thec-adic (resp. cp-adic) filtration on

K+/b · K+. The foregoing implies that the Frobenius endomorphism induces a morphism offiltered modulesFil•1(K

+/b · K+) → Fil•2(K+/b · K+) which is surjective on the associated

graded modules; by [11, Ch.III,§2, n.8, Cor.2] the claim follows.Next, assume (iv) and letW := (K+/p)q, for two consecutive prime idealsp ⊂ q ⊂ K+.

By assumption the Frobenius map is surjective onK+/b · K+, wheneverb ∈ K+ \ 0 and|b| ≥ |p|; we deduce easily that the Frobenius endomorphism is surjective onW/b ·W , whichimplies (iii), in view of proposition 5.6.6.

(i)⇒(iii): indeed, letp ⊂ q be as in (iii); we need to show that(K+/p)q is deeply ramified.After replacingK+ byK+

q we can assume thatq is the maximal ideal ofK+. The ringk+ :=K+/p is a valuation ring; letk := Frac(k+), | · |k the valuation onk corresponding tok+, and(k′, | · |k′) a finite separable valued extension of(k, | · |k). It suffices to show thatΩk′+/k+ = 0.We havek′ ≃ k[X ]/f(X) · k[X ] for some irreducible monic polynomialf(X) ∈ k[X ]; letf(X) ∈ K+

p [X ] be a lifting of f(X) to a monic polynomial. ThenE+ := K+p [X ]/f(X) ·

K+p [X ] is the integral closure ofK+ in the finite separable extensionE := Frac(E+) ofK, and

E+/p · E+ ≃ k′, so thatE+ is a valuation ring, by lemma 5.1.13. Furthermore, the preimageof k′+ in E+ is a valuation ringR of E with R ∩K = K+ andR/p · R ≃ k′+. From (i) andtheorem 5.3.28(ii) we deduce thatΩR/K+ = 0, whenceΩk′+/k+ = 0 as required.

Finally we show that (iv) implies (i). We distinguish several cases. The case whenp :=char(K) > 0 has already been dealt with, in view of remark 5.6.9. Next suppose thatchar(κ) =0; we will adapt the argument given for the rank one case to prove corollary 5.3.18(i). As usual,we reduce to the case whereK is strictly henselian; it suffices to show thatΩE+/K+ = 0 forevery finite extension(E, | · |E) of K. ThenE factors as a tower of subextensionsE0 := K ⊂

E1 ⊂ E2 ⊂ ... ⊂ En := E such thatEi+1 = Ei[b1/lii ] for every i = 0, ..., n − 1, where

li := [Ei+1 : Ei] is a prime number andbi ∈ Ei such that|bi| /∈ li · ΓEi. It is easy to see

that assumption (iv) is inherited by every finite algebraic extension ofK, hence we can reduceto the caseE = K[b1/l], with b := b1, l := l1. One verifies as in the proof of proposition5.3.11(i) thatE+ consists of the elements of the form

∑l−1i=0 xi · b

i/l such thatxi ∈ K and|xi · bi/l|E ≤ 1 for everyi = 0, ..., l − 1 and we have to show thatd(xi · bi/l) = 0 for everyi ≤ l − 1. We may assume thati > 0, and up to replacingb by bi · xli, we can obtain thatb ∈ E+; we have then to verify thatdb1/l = 0. Definep(b), q(b) as in lemma 5.6.10, so thatp(b) andq(b) are consecutive prime ideals, therefore(E+/p(b))q(b) is a deeply ramified rankone valuation ring; in particular, its value group is not discrete. Then, using (5.1.5.1.22) andexample 5.1.21(v), we deduce that there exists an elementc ∈ K+ such that|b| > |cl+1| and|b| < |cl|. We can writeb = x · cl for somex ∈ K+, whencedb1/l = c · dx1/l. However,|c| ≤ |b|(l−1)/l(l+1) ≤ |a1/l · b−1|l−1 = |x|(l−1)/l. Sincex(l−1)/l · dx1/l = 0, the claim follows.Finally, suppose thatp := char(κ) > 0 andchar(K) = 0. Arguing as in the previous case,we produce an elementb ∈ K+ such that|bp| > |p| and |bp+1| < |p|. The Frobenius map issurjective onK+/bp ·K+ by assumption, and onKs+/bp · Ks+ by example 5.1.21(iii), henceΩKs+/K+ ≃ ΩKs+[1/p]/K+[1/p], by lemma 5.5.9(ii). Now it suffices to remark thatK+[1/p] is avaluation ring with residue field of characteristic zero, sowe are reduced to the previous case,and the proof is concluded.


Remark 5.6.14. By inspection of the proof, it is easy to check that condition(ii) of theorem5.6.11 is equivalent to the following. There exists a subsetS ⊂ K+ \ 0 such that the convexsubgroup generated by|S| := |s| | s ∈ S equalsΓK andHi(LE+/K+ ⊗K+ K+/s ·K+) = 0for every valued field extension(E, | · |E) of (K, | · |), everys ∈ S and everyi > 0.


6. ANALYTIC GEOMETRY

Throughout this chapter we fix a valued field(K, | · |) with valuation of rank one, completefor its valuation topology. As usual,m denotes the maximal ideal ofK+. We also leta be atopologically nilpotent element inK×.

6.1. Derived completion functor. Let A be a completeK+-algebra of topologically finitepresentation. For anyA-moduleM , we denote byM∧ the (separated)a-adic completion ofM .

Proposition 6.1.1. LetA be as in(6.1).

(i) Every finitely generatedA-module which is torsion-free as aK+-module, is finitely pre-sented.

(ii) A is a coherent ring.(iii) LetN be a finitely presentedA-module,N ′ ⊂ N a submodule. Then there exists an integer

c ≥ 0 such that

ak ·N ∩N ′ ⊂ ak−c ·N ′(6.1.2)

for everyk ≥ c. In particular, the topology onN ′ induced by thea-adic topology onN ,agrees with thea-adic topology ofN ′.

(iv) Every finitely generatedA-module isa-adically complete and separated.(v) Every submodule of a freeA-moduleF of finite type is closed for thea-adic topology of

F .(vi) EveryA-algebra of topologically finite type is separated.

Proof. (i) is an easy consequence of [8, Lemma 1.2]. To show (ii), onechooses a presenta-tion A := K+〈T1, ..., Tn〉/I for some finitely generated idealI, and then reduces to provethe statement forK+〈T1, ..., Tn〉, in which case it follows from (i). Next, letN , N ′ be asin (iii) and defineT to be theK+-torsion submodule ofN ′′ := N/N ′; clearly T is anA-submodule, and theA-moduleN ′′/T isK+-torsion-free, therefore is finitely presented by (i).SinceN is finitely generated, this implies thatM := Ker(N → N ′′/T ) is finitely gener-ated. Hence, there exists an integerc ≥ 0 such thatac · M ⊂ N ′. If now k ≥ c, we haveak ·N ∩N ′ ⊂ ak ·N ∩M = ak ·M ⊂ ak−c ·N ′, which shows (iii). Next let us show:

Claim6.1.3. Assertion (iv) holds for every finitely presentedA-module.

Proof of the claim:Let N be a finitely presentedA-module and choose a presentation0 →K → An → N → 0. By (ii), K is again finitely presented, and by (iii), the topology onKinduced by thea-adic topology onAn coincides with thea-adic topology ofK. Hence, aftertakinga-adic completion, we obtain a short exact sequence :0 → K∧ → An → N∧ → 0 (see[41, Th.8.1]). It follows that the natural mapK → K∧ is injective, which shows that the mapN → N∧ is surjective for every finitely presentedA-moduleN . In particular, this holds forK,whenceK ≃ K∧, andN ≃ N∧, as claimed.

Finally, letM be a submodule ofAn. By (iii), the topology onM induced byAn coincideswith the a-adic topology. Consequently, ifM is finitely presented, thenM is complete forthea-adic topology by claim 6.1.3, hence complete as a subspace of An, hence closed inAn.For an arbitraryM , defineM :=

⋃n>0(M : an); thenM is a submodule ofAn andA/M is

torsion-free as aK+-module, so it is finitely presented by (i), thereforeM is finitely presentedby (ii). It follows thatac ·M ⊂ M for somec ≥ 0, whenceak · An ∩M = ak · An ∩M foreveryk ≥ c. By the foregoing,M is complete, soak ·An∩M is also complete, and finallyM iscomplete, hence closed. This settles (v) and (iv) follows aswell. Assertion (vi) is an immediateconsequence of (v).


Lemma 6.1.4. LetA→ B be a map ofK+-algebras of topologically finite presentation. ThenB is of topologically finite presentation as anA-algebra. More precisely, ifφ : A〈T1, ..., Tn〉 →B is any surjective map,Ker(φ) is finitely generated.

Proof. By proposition 6.1.1(vi),B is complete and separated, hence we can find a surjectivemapφ : A〈T1, ..., Tn〉 → B. It remains to show that for any suchφ,Ker(φ) is finitely generated.We can writeA := K+〈Tn+1, ..., Tm〉/I for some finitely generated idealI, and thus reduce tothe case whereA = K+ andφ : K+〈T1, ..., Tn〉 → B. We will need the following :

Claim 6.1.5. Let α : K+〈Y1, ..., Yr+s〉 → B be a surjective map andβ : K+〈Y1, ..., Yr〉 →K+〈Y1, ..., Yr+s〉 the natural imbedding. Suppose thatγ := α β is surjective as well. ThenKer(α) is finitely generated if and only ifKer(γ) is finitely generated.

Proof of the claim:For i = r+1, ..., r+ s, choosefi ∈ K+〈Y1, ..., Yr〉 such thatγ(fi) = α(Yi).We define a surjective mapδ : K+〈Y1, ..., Yr+s〉 → K+〈Y1, ..., Yr〉 by settingδ(Yi) := Yifor i ≤ r andδ(Yi) := fi for i > r. Clearlyγ δ = α. There follows a short exact sequence0→ Ker(δ)→ Ker(α)→ Ker(γ)→ 0. However,Ker(δ) is the closure of the idealI generatedby Yi − fi for i = r+ 1, ..., r+ s. By proposition 6.1.1(v), we deduce thatKer(δ) = I, and theclaim follows easily.

By hypothesis there is at least one surjectionψ : K+〈Y1, ..., Yr〉 → B with finitely gener-ated kernel. Letµ : B⊗K+B → B be the multiplication map and setθ := µ (φ⊗K+ψ) :K+〈T1, ..., Tn, X1, ..., Xk〉 → B. Applying twice claim 6.1 we deduce first thatKer(θ) isfinitely generated, and then thatKer(φ) is too, as required.

Lemma 6.1.6. LetF be a flatA-module. Then:

(i) F∧ is a flatA-module.(ii) For every finitely presentedA-moduleM , the natural map

M ⊗A F∧ → (M ⊗A F )

∧(6.1.7)

is an isomorphism.

Proof. To begin with, we claim that the functorN 7→ (N ⊗A F )∧ is exact on the abeliancategory of finitely presentedA-modules. Indeed, letE := (0 → N ′ → N → N ′′ → 0) be anexact sequence of finitely presentedA-modules; we have to show that(E ⊗A F )

∧ is still exact.ObviouslyE ⊗A F is exact, so the assertion will follow by [41, Th.8.1(ii)], once we know:

Claim 6.1.8. The topology onN ′⊗A F induced by the imbedding intoN ⊗A F agrees with thea-adic topology.

Proof of the claim:By proposition 6.1.1(iii), we can findc ≥ 0 such that (6.1.2) holds. SinceFis flat, we derive

ak(N ⊗A F ) ∩ (N ′ ⊗A F ) ⊂ ak−c(N ′ ⊗A F )

which implies the claim.

(ii): clearly (6.1.7) is an isomorphism in caseM is a free module of finite type. For a generalM , one chooses a resolutionR := (An → Am → M → 0); by the foregoing, the sequence(R ⊗A F )∧ is still exact, so one concludes by applying the 5-lemma to the map of complexesR ⊗A F∧ → (R⊗A F )∧.

(i): we have to show that, for every injective map ofA-modulesf : N ′ → N , f ⊗A 1F∧ isstill injective. By the usual reductions, we can assume thatbothN andN ′ are finitely presented.In view of (ii), this is equivalent to showing that the induced map(N ′ ⊗A F )∧ → (N ⊗A F )∧

is injective, which is already known.


6.1.6.1.9. We will need to consider the left derived functorof thea-adic completion functor,which we denote:

D−(A-Mod)→ D−(A-Mod) : (K•) 7→ (K•)∧.(6.1.10)

As usual, it can be defined by completing termwise bounded above complexes of projectiveA-modules. However, the following lemma shows that it can also be computed by arbitrary flatresolutions.

Lemma 6.1.11.Letφ : K•1 → K•2 be a quasi-isomorphism of bounded above complexes of flatA-modules and denote by(K•i )

∧ the termwisea-adic completion ofK•i (i = 1, 2). Then theinduced morphism

(K•1 )∧ → (K•2)

∧(6.1.12)

is a quasi-isomorphism.

Proof. Since(K•)1 and(K•)2 are termwise flat, we deduce quasi-isomorphisms

φn : K•1,n := K•1 ⊗A A/an ·A→ K2,n := K•2 ⊗A A/a

n · A

for every n ∈ N. The map of inverse system of complexes(K•1,n)n∈N → (K•2,n)n∈N canbe viewed as a morphism of complexes of objects of the abeliancategory(A-Mod)N of in-verse systems ofA-modules. As such, it induces a morphism(φn)n∈N in the derived categoryD((A-Mod)N), and it is clear that(φn)n∈N is a quasi-isomorphism. Let

R lim : D((A-Mod)N)→ D(A-Mod)

be the right derived functor of the inverse limit functorlim : (A-Mod)N → A-Mod. Weremark that, for everyj ∈ Z, the inverse systems(Kj

i,n)n∈N (i = 1, 2) are acyclic for the functorlim, since their transition maps are surjective. We derive thatR lim(K•i,n)n∈N ≃ (K•i )

∧, and,under this identification, the morphism (6.1.12) is the sameasR lim(φn)n∈N. Since the latterpreserves quasi-isomorphisms, the claim follows.

6.1.6.1.13. We denote byD−(A-Mod)∧ the essential image of the functor (6.1.10).

Corollary 6.1.14. (i) For any objectK• of D−(A-Mod), the natural morphism

(K•)∧ → ((K•)∧)∧


(ii) D−(A-Mod)∧ is a full triangulated subcategory ofD−(A-Mod).

Proof. Notice first that there are two natural morphisms as in (i), which coincide : namely,for any complexE in D−(A-Mod) one has a natural morphismuE : E → E∧; then onecan take either(uK)∧ or uK∧. Now, (i) is an immediate consequence of lemmata 6.1.11 and6.1.6. ClearlyD−(A-Mod)∧ is preserved by shift and by taking cones of arbitrary morphisms;furthermore, it follows from (i) that it is a full subcategory of D−(A-Mod).

We will need some generalities on pseudo-coherent complexes ofR-modules (for an arbitraryring R), which we borrow from [6, Exp.I]. In our situation, the definitions can be simplifiedsomewhat, since we are only concerned with sheaves over the one-point site that are pseudo-coherent relative to the subcategory of freeA-modules of finite type.

6.1.6.1.15. For givenn ∈ Z, one says that a complexK• of R-modules isn-pseudo-coherentif there exists a quasi-isomorphismE• → K• whereE• is a complex bounded above such thatEi is a freeR-module of finite type for everyi ≥ n. One says thatK• is pseudo-coherentif itis n-pseudo-coherent for everyn ∈ Z.


6.1.6.1.16. LetK• be an-pseudo-coherent (resp. pseudo-coherent) complex ofR-modules,andF • → K• a quasi-isomorphism. ThenF • is n-pseudo-coherent (resp. pseudo-coherent)([6, Exp.I, Prop.2.2(b)]). It follows that that the pseudo-coherent complexes form a (full) sub-categoryD(R-Mod)coh of D(R-Mod).

6.1.6.1.17. Furthermore, letX → Y → Z → X [1] be a distinguished triangle inD−(R-Mod).If X andZ aren-pseudo-coherent (resp. pseudo-coherent), then the same holds forY ([6, Exp.I,Prop.2.5(b)]).

Lemma 6.1.18.Let n, p ∈ N, K• a n-pseudo-coherent complexes inD≤0(R-Mod), andFp

one of the functors⊗pR, SympR, ΛpR, ΓpR defined in[34, I.4.2.2.6]. ThenLFp(K

•) is an n-pseudo-coherent complex.

Proof. It is well known thatFp sends freeR-modules of finite type to freeR-modules of finitetype. It follows easily that the assertion of the lemma can bechecked by inspecting the definitionof the unnormalized chain complex associated to a simplicial complex, and of the simplicialcomplex associated to a chain complex via the Dold-Kan correspondence. We omit the details.

6.1.6.1.19. LetK• be a pseudo-coherent complex. By ([6, Exp.I, Prop.2.7]) there exists aquasi-isomorphismE• → K• whereE• is a bounded above complex of freeR-modules offinite type.

6.1.6.1.20. Suppose now thatR is coherent; then we deduce easily that a complexK• of R-modules is pseudo-coherent if and only ifH i(K•) is a coherentR-module for everyi ∈ Z andH i(K•) = 0 for every sufficiently largei ∈ Z ([6, Exp.I, Cor.3.5]). By proposition 6.1.1(iv) itfollows also thatD−(A-Mod)coh ⊂ D−(A-Mod)∧ for everyK+-algebraA as in (6.1).

6.1.6.1.21. LetA be as in (6.1) andM anA-module of finite presentation. We denote byM [0]the complex consisting of the moduleM placed in degree zero. Any finite presentation ofMcan be extended to a quasi-isomorphismE• → M [0], whereE• is a complex of freeA-modulesof finite type andEi = 0 for i > 0 ([6, Exp.I, Cor.3.5(a)]). Together with proposition 6.1.1(iv),it follows easily that the natural morphismM [0]→ M [0]∧ is a quasi-isomorphism.

Lemma 6.1.22.LetA→ B be a map of completeK+-algebras of topologically finite presen-tation. Then, for every objectK• of D−(A-Mod), the natural morphism

(K•L

⊗A B)∧ → (K•∧L

⊗A B)∧


Proof. We can suppose thatK• is a complex of freeA-modules. Then we are reduced toshowing that, for every freeA-moduleF , the natural map(F ⊗A B)∧ → (F∧ ⊗A B)∧ is anisomorphism. We leave this task to the reader.

Definition 6.1.23. Let φ : A → B be a map of completeK+-algebras of topologically finitepresentation. Theanalytic cotangent complexof φ is the complexLan

B/A := (LB/A)∧. Notice thatLB/A is defined here via the standard resolutionPA(B) → B, and it is therefore well definedas acomplexof B-modules, not just as an object in the derived categoryD−(B-Mod)∧. Thiswill be essential in order to globalize the construction to formal schemes, in (6.2.6.2.2), and toZariski-Riemann spaces, in (6.2.6.2.21).


Proposition 6.1.24.Letφ : A → B be a map of completeK+-algebras of topologically finitepresentation, and suppose thatφ is formally smooth for thea-adic topology. Then there is anatural quasi-isomorphism

LanB/A ≃ ΩB/A[0]

∧.

Proof. For everyn ∈ N, setAn := A/an ·A andBn := B/an ·B. The hypothesis onφ impliesthatφn := φ ⊗A 1An is of finite presentation and formally smooth for the discrete topology,therefore

LBn/An ≃ ΩBn/An[0] ≃ ΩB/A ⊗A An[0](6.1.25)

for everyn ∈ N. Moreover,φ is flat by [8, Lemma 1.6], henceLBn/An ≃ LB/A ⊗A An. On theother hand, for everyi ∈ Z there is a short exact sequence (cp. [50, Th.3.5.8])

0→ limn∈N

1H i−1(LB/A ⊗A An)→ H i(LanB/A)→ lim

n∈NH i(LB/A ⊗A An)→ 0.

In view of (6.1.25), the inverse system(H i−1(LB/A ⊗A An))n∈N vanishes fori 6= 1 and hassurjective transition maps fori = 1, hence itslim1 vanishes for everyi ∈ Z, and the claimfollows easily.

Proposition 6.1.26.Let φ : A → B be a surjective map of completeK+-algebras of topo-logically finite presentation. ThenLB/A is a pseudo-coherent complex, in particular it lies inD−(B-Mod)∧ andLB/A ≃ Lan

B/A.

Proof. First of all, notice that by lemma 6.1.4,B is of finite presentation, hence it is coherent asanA-module. LetP := PA(B) be the standard simplicial resolution ofB by freeA-algebras.We obtain a morphism of simplicialB-algebrasφ : B ⊗A P → B by tensoring withB theaugmentationP → B (hereB is regarded as a constant simplicial algebra). By the foregoing,P is pseudo-coherent, henceP ⊗A B lies in D(B-Mod)coh. Let J := Ker(φ). The short exactsequence of complexes0 → J → P ⊗A B → B → 0 is split, thereforeJ is also pseudo-coherent. Recall that we have natural isomorphisms:J i/J i+1 ∼→ Symi

B(LB/A) for everyi ∈ N(whereJ0 := B ⊗A P andSym0

B(LB/A) := B) ([34, Ch.III,§3.3]). Furthermore, we have (seeloc.cit.) :

Hn(Ji) = 0 for everyn, i ∈ N such thati > n.(6.1.27)

We prove by induction onn thatLB/A is n-pseudo-coherent for everyn ≤ 1. If n = 1 there isnothing to prove. Suppose that the claim is known for the integern. It then follows by lemma6.1.18 thatJ i/J i+1 isn-pseudo-coherent for everyi > 0. However, it follows from (6.1.27) thatJ i is n-pseudo-coherent as soon asi > −n. Hence, by (6.1.6.1.17) (and an easy induction), wededuce thatJ i isn-pseudo-coherent for everyi ∈ N. HenceJ2[1] is (n−1)-pseudo-coherent; ifwe now apply (6.1.6.1.17) to the distinguished triangleJ → LB/A → J2[1]→ J [1], we deducethatLB/A is (n− 1)-pseudo-coherent.

Theorem 6.1.28.LetA → B → C be maps of completeK+-algebras of topologically finitepresentation. Then:

(i) LanB/A lies inD−(B-Mod)coh.

(ii) There is a natural distinguished triangle inD−(C-Mod) :

C ⊗B LanB/A → Lan

C/A → LanC/B → C ⊗B Lan

B/A[1].(6.1.29)

Proof. (i): by lemma 6.1.4 we can find a surjectionB0 := A〈T1, ..., Tn〉 → B from a topolog-ically freeA-algebra ontoB. If we apply transitivity to the sequence of mapsA → B0 → B


and take the (derived) completion of the resulting distinguished triangle, we end up with thetriangle:

(B ⊗B0 LB0/A)∧ → Lan

B/A → LanB/B0

→ (B ⊗B0 LB0/A)∧[1].

We know already from proposition 6.1.26 thatLB/B0is pseudo-coherent, hence it coincides with

LanB/B0

. Lemma 6.1.22 yields a quasi-isomorphism:(B⊗B0LB0/A)∧ ∼→ (B⊗B0L

anB0/A

)∧; in viewof proposition 6.1.24,Lan

B0/Ais a freeB0-module of finite rank in degree zero, in particular it is

pseudo-coherent, so the same holds for(B ⊗B0 LB0/A)∧, and taking into account (6.1.6.1.17),

the claim follows.(ii): if we apply transitivity to the sequence of mapsA→ B → C, and then we complete the

distinguished triangle thus obtained, we obtain (6.1.29),except that the first term is replaced by(C ⊗B LB/A)∧, which we can also write as(C ⊗B Lan

B/A)∧, in view of lemma 6.1.22. However,

by (i),LanB/A is pseudo-coherent, so it remains such after tensoring byC; in particularC⊗BLan

B/A

is already complete, and the claim follows.

6.2. Cotangent complex for formal schemes and adic spaces.In this section we show howto globalize the definition of the analytic cotangent complex introduced in section 6.1. Weconsider two kinds of globalization : first we define the cotangent complex of a morphismf : X → Y of formal schemes locally of finite presentation overSpf(K+); then we will definethe cotangent complex for the morphism of Zariski-Riemann spaces associated tof.

Lemma 6.2.1. LetX := Spf(A) be an affine formal scheme finitely presented overSpf(K+).For everyf ∈ A, letD(f) := x ∈ X | f /∈ mx. The natural mapA→ Γ(D(f),OX) is flat.

Proof. SinceΓ(D(f),OX) is the a-adic completion ofAf , the lemma follows from lemma6.1.6(i).

6.2.6.2.2. Letf : X→ Y be a morphism of formal schemes locally of finite presentation overSpf(K+), and suppose thatY is separated. For every affine open subsetU ⊂ X, the smallcategoryFU of all affine open subsetsV ⊂ Y with f(U) ⊂ V , is cofiltered under inclusion (orelse it is empty). For everyV ∈ FU , OY(V ) is aK+-algebra of topologically finite presentation,hence the induced morphismOY(V ) → OX(U) is of the kind considered in definition 6.1.23.We set

L(U/Y) := colimV ∈F o

U

LanOX (U)/OY (V ).

Definition 6.2.3. The mappingU 7→ L(U/Y) defines a complex of presheaves on a cofinalfamily of affine open subsets ofX. By applying degreewise the construction of [18, Ch.0,§3.2.1], we can extend the latter to a complex of presheaves ofOX-modules onX. We definethecotangent complexLX/Y of the morphismf : X→ Y as the complex of sheaves associatedto this complex of presheaves (this means that we form degreewise the associated sheaf, and weconsider the resulting complex).

6.2.6.2.4. More generally, ifY is not necessarily separated, we can choose an affinoid cover-ing Y =

⋃i∈I Ui and the construction above applies to the restrictionsVi := f−1(Ui) → Ui;

since the definition ofLVi/Uiis local onVi, one can then glue them into a single cotangent

complexLX/Y.

Proposition 6.2.5. For everyf : X → Y as in (6.2.6.2.2), LX/Y is a pseudo-coherent complexof OX-modules.

Proof. According to [6, Exp.I, Prop.2.1(b)], it suffices to show thatHi(LX/Y) is a coherent sheafof OX-modules for everyi ∈ N. To this aim, letU ⊂ X be an affine open subset such that thefamily FU (notation of (6.2.6.2.2)) is not empty; pick anyV ∈ FU . After replacingX byU , wecan suppose thatU = X. SetA := Γ(X,OX) and letLi be the sheaf of coherentOX-modules


associated to the coherentA-moduleLi := Hi(LanOX (X)/OY (V )) (cp. [18, Ch.I,§10.10.1], where

this concept is discussed in the case of locally noetherian formal schemes). The assertion willbe an immediate consequence of theorem 6.1.28(i) and the following :

Claim6.2.6. There is a natural isomorphism ofOX-modules:Li∼→ Hi(LX/Y).

Proof of the claim:By the definition ofLX/Y we deduce a natural morphism ofOX-modules:α : Li → Hi(LX/Y). It therefore suffices to show thatα induces an isomorphism on the stalks.To this aim, we remark first that the natural mapLi → Hi(L(X/Y)) is an isomorphism. Indeed,it suffices to consider another open subsetV ′ ∈ FX with V ′ ⊂ V ; we haveLan

OY (V ′)/OY (V ) ≃0 by proposition 6.1.24, and then it follows by transitivity (theorem 6.1.28(ii)) that the mapLi → Hi(Lan

OX (X)/OY (V ′)) is an isomorphism. More generally, this argument shows that, forevery affine open subsetU ′ ⊂ X, the natural mapHi(Lan

OX (U ′)/OY (V )) → Hi(L(U′/Y)) is an

isomorphism. However, on one hand we have(Li )x ≃ Li ⊗A OX,x. On the other hand, wehave ([18, Ch.0,§3.2.4]) :

Hi(LX/Y)x ≃ colimx∈U ′

Hi(L(U′/Y))(6.2.7)

where the colimit ranges over the setS of all affine open neighborhoods ofx in X. We canreplaceS by the cofinal subset of all open neighborhoods of the formD(f) (for everyf ∈ Asuch thatf /∈ mx). Then, lemma 6.2.1, together with another an easy application of transitivityallows to identify the right-hand side of (6.2.7) withHi(L(X/Y))⊗AOX,x, so the claim follows.

Proposition 6.2.8. Let Xf→ Y

g→ Z be two morphisms of formal schemes locally of finite

presentation overSpf(K+). There is a natural distinguished triangle inD−(OX-Mod)

Lf∗LY/Z→ LX/Z→ LX/Y→ Lf∗LY/Z[1].(6.2.9)

Proof. As explained in [34, Ch.II,§2.1], for every sequence of ring homomorphismsA→ B →C, the transitivity triangle is induced by a functorial exactsequence of complexesLC/B/A of flatC-modules (a ”true triangle” inloc. cit.). Suppose now thatA,B,C are completeK+-algebrasof topologically finite type; upona-adic completion, one deduces a true triangleL∧C/B/A. Then,to every sequence of affine open subsetsU ⊂ X, V ⊂ Y, W ⊂ Z such thatf(U) ⊂ V andg(V ) ⊂ W , one can associate the true triangleL∧

OX (U)/OY (V )/OZ(W ); Since the construction isfunctorial in all arguments, one derives a presheaf of true triangles on a cofinal family of opensubsets ofX, which we can then sheafify in the usual manner. The resultingdistinguishedtriangle inD−(OX-Mod) gives rise to (6.2.9).

In the following we wish to define a cotangent complex for the morphism of Zariski-Riemannspaces associated to a morphism of formal schemes. We take the viewpoint according to whichZariski-Riemann spaces are special cases of adic spaces as studied in [32] and [33]. For theconvenience of the reader we recall a few basic definitions from [33].

6.2.6.2.10. Anf-adic ring is a topological ringA that admits an open subringA0 such that theinduced topology onA0 is pre-adic and defined by a finitely generated idealI ⊂ A0. As anexample, everyK-algebra of topologically finite type is an f-adic ring. A subringA0 with theabove properties is called aring of definitionfor A, andI is anideal of definition. One denotesbyA the open subring of power-bounded elements ofA.

6.2.6.2.11. LetA→ B be complete f-adic rings andφ : A→ B a ring homomorphism. Onesays thatφ is of topologically finite typeif there exist rings of definitionA0 ⊂ A andB0 ⊂ Bsuch thatφ(A0) ⊂ B0, the restrictionA0 → B0 factors through a quotient map (i.e. open andsurjective)A0〈T1, ..., Tn〉 → B0 andB is finitely generated overA · B0.


6.2.6.2.12. Anaffinoid ring is a pairA = (A⊲, A+) consisting of an f-adic ringA⊲ and asubringA+ ⊂ A⊲ which is open, integrally closed inA⊲ and contained in the subringA. A+ iscalled thesubring of integral elementsof A.

6.2.6.2.13. ThecompletionA∧ of an affinoid ringA = (A⊲, A+) is the pair((A⊲)∧, (A+)∧) (itturns out that(A+)∧ is integrally closed in(A⊲)∧).

A homomorphismφ : (A⊲, A+) → (B⊲, B+) of affinoid rings is a ring homomorphismφ⊲ : A⊲ → B⊲ such thatφ(A+) ⊂ B+. One says thatφ is of topologically finite typeif φ⊲ is oftopologically finite type and there exists an open subringC ⊂ B+ such thatB+ is the integralclosure ofC, φ(A+) ⊂ C and the induced mapA+ → C is of topologically finite type. (cp.(6.2.6.2.11)).

6.2.6.2.14. Given an arbitrary ringA, avaluationonA is a map| · | : A→ Γ∪0 whereΓ isan ordered abelian group whose composition law we denote multiplicatively, and the orderingis extended toΓ ∪ 0 as usual. Then| · | is required to satisfy the usual conditions, namely:|x · y| = |x| · |y| and|x+ y| ≤ max(|x|, |y|) for everyx, y ∈ A, and|0| = 0, |1| = 1.

6.2.6.2.15. Now, letA be an f-adic ring, and| · | : A → Γ ∪ 0 a valuation onA. For everyγ ∈ Γ, letUγ := α ∈ Γ | α < γ ∪ 0. We endowΓ ∪ 0 with the topology which restrictsto the discrete topology onΓ, and which admits(Uγ | γ ∈ Γ) as a fundamental system of openneighborhoods of0. We say that| · | is continuousif it is continuous with respect to the abovetopology onΓ∪0. One denotes byCont(A) the set of all (equivalence classes of) continuousvaluations onA. Givena, b ∈ A, let U(a/b) ⊂ Cont(A) be the subset of all valuations| · |such that|a| ≤ |b| 6= 0. Cont(A) is endowed with the topology which admits the collection(U(a/b) | a, b ∈ A) as a sub-basis. With this topology,Cont(A) is a spectral topological space(see [33, 1.1.13] for the definition of spectral space). In particular, this implies thatCont(A)admits a basis of quasi-compact open subsets. Such a basis isprovided by therational subsets,defined as follows. A subsetU ⊂ Cont(A) is called rational if there existf1, ..., fn, g ∈ A suchthat the idealJ := f1 ·A+ ...+fn ·A is open inA andU consists of all| · | ∈ Cont(A) such that|fi| ≤ |g| 6= 0 for everyi = 1, ..., n. (Notice that, since we have chosen to restrict to f-adic ringscontainingK, asking forJ to be an open ideal is the same as requiring thatJ = A). Givenf1, ..., fn, g ∈ A with the above property, we denote byR(f1/g, ..., fn/g) the correspondingrational subset.

6.2.6.2.16. IfA := (A⊲, A+), then one defines the subsetSpa(A) := | · | ∈ Cont(A⊲) | |a| ≤1 for everya ∈ A+ ⊂ Cont(A⊲). Spa(A), endowed with the subspace topology, is called theadic spectrumof the affinoid ringA. Spa(A) is a pro-constructible subset ofCont(A), henceit is a spectral space too. Any continuous mapA → B of affinoid rings induces in the obviousway a continuous map on adic spectra:Spa(B)→ Spa(A).

6.2.6.2.17. For any affinoid ringA, one can endowX := Spa(A) with a presheafOX of topo-logical rings, as follows. First of all, for anyf1, ..., fn, g ∈ A⊲ as in (6.2.6.2.15), one defines anaffinoid ringA(f1/g, ..., fn/g), such thatA(f1/g, ..., fn/g)⊲ := (A⊲)g andA(f1/g, ..., fn/g)+

is the integral closure of the subringA[f1/g, ..., fn/g] in A(f1/g, ..., fn/g)⊲. If B ⊂ A⊲ is a

ring of definition andI ⊂ B an ideal of definition, letB(f1/g, ..., fn/g) be the subring of(A⊲)ggenerated byB andf1/g ,..., fn/g; we endowB(f1/g, ..., fn/g) with the pre-adic topologydefined by the idealI · B(f1/g, ..., fn/g); then the f-adic topology onA(f1/g, ..., fn/g)⊲ is de-fined to be the unique ring topology for whichB(f1/g, ..., fn/g) is a ring of definition. Next, letA〈f1/g, ..., fn/g〉 := A(f1/g, ..., fn/g)

∧ (cp. (6.2.6.2.13)). With this preliminaries, one sets:

OX(R(f1/g, ..., fn/g)) := A〈f1/g, ..., fn/g〉⊲.

In this way,OX is well defined on every rational subset. One can then extend the definition toan arbitrary open subset ofSpa(A), following [18, Ch.0,§3.2.1]. It is not difficult to check that,


for every open subsetU ⊂ Spa(A), and everyx ∈ U , any valuation| · |x in the equivalence classx extends to the whole ofOX(U), hence to the stalkOX,x. One denotes byO+

X the sub-presheafdefined by the rule:O+

X(U) := f ∈ OX(U) | |f |x ≤ 1 for everyx ∈ U. In the cases ofinterest, the presheafOX is a sheaf (andO+

X is therefore a subsheaf). In such cases, one canshow that, for every rational subsetR(f1/g, ..., fn/g), the natural mapA〈f1/g, ..., fn/g〉+ →O

+X(R(f1/g, ..., fn/g)) is an isomorphism of topological rings.This holds notably whenA⊲ is aK-algebra of topologically finite type. One calls the datum

(Spa(A),OSpa(A),O+Spa(A)) anaffinoid adic space. General adic spaces are obtained as usual, by

gluing affinoids. Adic spaces form a category, whose morphismsf : X → Y are the morphismsof topologically locally ringed spaces(X,OX)→ (Y,OY ) which induce morphisms of sheavesf ∗O+

Y → O+X .

6.2.6.2.18. Letf : X → Y be a morphism of adic spaces. One says thatf is locally of finitetypeif for every x ∈ X there exist open affinoid subspacesU ⊂ X, V ⊂ Y such thatx ∈ U ,f(U) ⊂ V and the induced morphism of affinoid rings(OY (V ),O+

Y (V ))→ (OX(U),O+X(U))

is of topologically finite type.

6.2.6.2.19. A morphismf : X → Y between adic spaces (defined overSpa(K,K+)) iscalledsmooth(resp.unramified, resp.etale) if f is locally of finite type and if, for any affinoidring A, any idealI of A⊲ with I2 = 0 and any morphismSpa(A) → Y , the mappingHomY (Spa(A), X)→ HomY (Spa(A/I), X) is surjective (resp. injective, resp. bijective).

6.2.6.2.20. In [33,§1.9] it is shown how to associate functorially to every formal schemeX(say locally of finite presentation overSpf(K+)) an adic spaced(X), together with a morphismof topologically ringed spacesλ : d(X) → X, characterized by a certain universal propertywhich we won’t spell out here, but that includes the condition thatIm(OX→ λ∗Od(X)) ⊂ O

+d(X).

If X = Spf(A0) for aK+-algebraA0 of topologically finite type, thend(X) = Spa(A), whereA is the affinoid ring(A0 ⊗K+ K,A+), withA+ defined as the integral closure of the image ofA0 in A0 ⊗K+ K. Moreover,X is quasi-compact if and only ifd(X) is.

6.2.6.2.21. LetX be a formal scheme of finite presentation overSpf(K+). The collectionCX

of all morphismsf : X′ → X of formal schemes of finite presentation overSpf(K+) such thatd(f) is an isomorphism, forms a small cofiltered category (with morphisms given as usual by thecommutative diagrams). It is shown in [32] that there is a natural isomorphism of topologicallyringed spaces

(d(X),O+d(X))

∼→ lim

(X′→X)∈C

(X′,OX′).(6.2.22)

(Actually, the argument inloc.cit. is worked out only in the case of noetherian formal schemes,but it is not difficult to adapt it to the present situation). This leads us to make the following:

Definition 6.2.23. For every morphismf : X → Y of formal schemes of finite presentationover Spf(K+), let Cf be the category of all commutative diagrams of formal schemes overSpf(K+) of the kind:

X′ //

g1

Y′

g2

X

f // Y

such thatd(g1) andd(g2) are isomorphisms. Furthermore, for everyX′ ∈ CX, let πX′ : d(X)→X′ be the natural morphism of topologically ringed spaces. Thecotangent complexof the in-duced morphism of adic spaced(X)→ d(Y) is the complex

L+d(X)/d(Y) := colim

(X′→Y′)∈Cf

π∗X′LX′/Y′ .


We also setLd(X)/d(Y) := L+d(X)/d(Y) ⊗K+ K.

6.2.6.2.24. LetA, B be complete f-adic rings andA→ B a ring homomorphism of topologi-cally finite presentation. We refer to [33,§1.6] for the construction of auniversalA-derivationof B, which is a continuousA-derivationd : B → Ωan

B/A from B to a complete topologicalB-moduleΩan

B/A, universal forA-derivationsB → M to complete topologicalB-modulesM .The construction ofΩan

B/A can be globalized easily to various contexts : one obtainse.g.a sheafof relative differentialsΩX/Y (resp.ΩX/Y ) for any morphism of formal schemesX→ Y (resp.of adic spacesX → Y ) locally of finite type. Then one checks easily that:

H0(LX/Y) ≃ ΩX/Y(6.2.25)

and similarly for the adic variantLd(X)/d(Y).

Theorem 6.2.26.Let f : X→ Y be a morphism of formal schemes locally of finite presentationoverSpf(K+) and suppose that the induced morphismd(f) : d(X)→ d(Y) is smooth. Then:

(i) LX/Y⊗K+ K ≃ ΩX/Y[0]⊗K+ K in D−((OX⊗K+ K)-Mod) and(ii) Ld(X)/d(Y) ≃ Ωd(X)/d(Y)[0] in D−(Od(X)-Mod).

Proof. (i): after the usual reductions, we come to the following situation. We have a map ofK+-algebras of topologically finite presentationφ : A0 → B0, such thatd(Spf(φ)) is a smoothmorphism of affinoid adic spaces. We have to show thatLan

B0/A0⊗K+ K ≃ Ωan

B0/A0⊗K+ K[0]

in D(B-Mod). We can writeB0 = C0/I0, whereC0 := A0〈T1, ..., Tn〉 andI0 is some finitelygenerated ideal. SetI := I0 ⊗K+ K, A := A0 ⊗K+ K and letn be a maximal ideal ofC := C0 ⊗K+ K with I ⊂ n.

Claim 6.2.27. In is generated by a regular sequence of elements of the local ringCn.

Proof of the claim:Let p := n ∩A; p is a maximal ideal inA, and its residue fieldK ′ is a finiteextension ofK. Let n be the image ofn in C ⊗A K ′ andI ⊂ Od(Spf(C0)) the sheaf of idealscorresponding toI; the maximal idealn yields a point ind(Spf(C0)), which we denote byx(n).We have an isomorphism on then-adic completions:

(Ix(n))∧ ≃ (In)

∧.(6.2.28)

Moreover, there are natural maps :

Ix(n)/I2x(n) → n/n2 → Ωan

C/A ⊗C C/n

and, by [9, Prop.2.5], there exists a set of generatorsg1, ..., gk for Ix(n) such that the imagesdg1, ..., dgk in Ωan

C/A ⊗C C/n are linearly independent; it follows that the imagesg1, ..., gk in

n/n2 are also linearly independent. Due to (6.2.28), we can assume thatg1, ..., gk ∈ In, and thenit follows thatg1 ⊗ 1, ..., gk ⊗ 1 are the firstk elements of a regular system of parameters forthe regular local ringCn⊗A K ′. From lemma 6.1.6 it follows thatCn is a flatAp-module; thenby [20, Ch.0, Prop.15.1.16] we deduce thatg1, ..., gn is a regular sequence of elements ofCn, asrequired.

SetB := B0 ⊗K+ K; it follows from claim 6.2.27 and [34, Ch.III, Prop.3.2.4] thatCn ⊗CLC/B ≃ LCn/Bn ≃ In/I

2n[1] for every maximal idealn, henceLC/B ≃ I/I2[1]. By proposition

6.1.26, we deriveLanC0/B0

⊗K+ K ≃ I/I2[1]. Finally, by theorem 6.1.28(ii) and proposition

6.1.24 we deduce an isomorphism inD−(B-Mod) :

LanB0/A0

⊗K+ K ≃ (0→ I/I2 → B ⊗C0 Ω∧C0/A0

→ 0)

and the latter complex is quasi-isomorphic toΩanB/A[0] by [9, Prop.2.5].

Assertion (ii) is a consequence of (i) and of the definition ofLd(X)/d(Y).


Lemma 6.2.29.LetA → B be a continuous map ofK+-algebras of topologically finite pre-sentation. The universal property of(6.2.6.2.24)gives a natural map ofB-modules

φB/A : ΩB/A → ΩanB/A.

The mapφB/A is surjective withK+-divisible kernel.

Proof. One writesB = B0/I with B0 := A〈T1, ..., Tn〉 andI ⊂ B0. Directly from the con-struction ofΩan

B0/Aone checks thatφB0/A is onto. Then, a little diagram chasing shows that

φB/A is onto as well, and yields a surjective mapKer(φB0/A)⊗B0 B → Ker(φB/A). This allowsto reduce to the case whereB = B0. In this case,Ker(φB/A) is generated by the terms of theform δ(f) := df −

∑ni=1(∂f/∂Ti) · dTi, wheref ranges over all the elements ofB0. For given

f ∈ B0, we can writef = f0 + a · f1, with f0 ∈ A[T1, ..., Tn], f1 ∈ B0. It follows easily thatδ(f) = δ(a · f1) = a · δ(f1), whence the claim.

Remark 6.2.30. In view of (6.2.25), lemma 6.2.29 is also implied by the following more gen-

eral observation. LetKB/A := Cone(ψB/A : LB/A → LanB/A)[1]; one has:KB/A

L

⊗B B/aB ≃ 0.

Indeed, directly on the definition ofLanB/A one sees thatψB/A

L

⊗B 1B/aB is an isomorphism.

6.3. Analytic geometry over a deeply ramified base.In this section we assume throughoutthat(K, | · |) is a deeply ramified complete valued field, with valuation of rank one. Recall thata ∈ K× denotes a topologically nilpotent element ofK.

If X is a formal scheme of finite type overSpf(K+), we will sometimes writeLX/K+ insteadof LX/Spf(K+). Similarly we defineL+

d(X)/K and setΩ+d(X)/K := H0(L

+d(X)/K).

Theorem 6.3.1.LetX be a formal scheme of finite type overSpf(K+) such thatX := d(X) isa smooth adic space overSpa(K,K+). ThenL+

X/K ≃ Ω+X/K [0] in D−(O+-Mod), andΩ+

X/K

is a flat sheaf ofO+X-modules.

Proof. Both assertions can be checked on the stalks, therefore letx ∈ X be any point. The stalkOX,x is a local ring and its residue fieldκ(x) carries a natural valuation; the preimage inOX,x ofthe corresponding valuation ringκ(x)+ is the subringO+

X,x. Let I :=⋂n∈N a

nO+X,x; it follows

from this description thatκ(x)+ = O+X,x/I. Especially, we have

O+X,x/a · O

+X,x ≃ κ(x)+/a · κ(x)+.(6.3.2)

Claim6.3.3. LetM be anO+X,x-module, and suppose thata is regular onM . ThenM/IM is a

flat κ(x)+-module.

Proof of the claim:By snake lemma we deriveKer(M/IM·a→M/IM) ⊂ Coker(IM

·a→ IM).

However, it is clear thatI = a · I, soa is regular onM/IM , whence the latter is a torsion-freeκ(x)+-module and the claim follows.

Let U be the cofiltered system of all affinoid open neighborhoods ofx in X; for U ∈ U letFU be the filtered system of allK+-subalgebras ofO+

X(U) of topologically finite presentation.We derive

(L+X/K)x

L

⊗K+ K+/aK+ ≃ colimU∈U

colimA∈FU

LanA/K+

L

⊗K+ K+/aK+

≃ colimU∈U

colimA∈FU

LA/K+

L

⊗K+ K+/aK+

≃ LO

+X,x/K

+

L

⊗K+ K+/aK+

≃ Lκ(x)+/K+

L

⊗K+ K+/aK+.


Together with theorem 5.6.11, this implies already that scalar multiplication bya is an auto-morphism ofHi(L

+X/K), for everyi > 0. However, according to theorem 6.2.26(ii),Hi(L

+X/K)

is aK+-torsion sheaf ofO+X-modules, fori > 0, whence the first assertion. It also follows that

(Ω+X/K)x is a torsion-free, hence flat,K+-module. To prove that(Ω+

X/K)x is a flatO+X,x-module,

we remark first that(ΩX/K)x ≃ (Ω+X/K)x ⊗K+ K, and the latter is a flatOX,x-module, sinceX

is smooth overSpa(K,K+). By [31, Partie II, lemma 1.4.2.1] it suffices therefore to show

Claim 6.3.4. (Ω+X/K)x ⊗K+ K+/aK+ is a flatO+

X,x ⊗K+ K+/aK+-module.

Proof of the claim:By claim 6.3.3 we know that(Ω+X/K)x ⊗O

+X,x

κ(x)+ is a flatκ(x)+-module.

In view of (6.3.2), the claim follows after base change toκ(x)+/a · κ(x)+.

Definition 6.3.5. Let (Xα | α ∈ I) be a system of formal schemes of finite presentation overSpf(K+), indexed by a small cofiltered categoryI.

(i) Let X∞ := limα∈I

Xα, where the limit is taken in the category of topologically ringed spaces.

For everyα ∈ I, letπα : X∞ → Xα be the natural morphism of locally ringed spaces. WedefineΩX∞/K+ := colim

α∈Ioπ∗α(ΩXα/K+), which is a sheaf ofOX∞-modules.

More generally, we letLX∞/K+ := colimα∈Io

π∗α(LXα/K+).

(ii) We say that the cofiltered system(Xα | α ∈ I) is deeply ramifiedif the natural morphismΩX∞/K+ → ΩX∞/K+ ⊗K+ K is an epimorphism.

Lemma 6.3.6. Let (Xα | α ∈ I) be a cofiltered system as in definition 6.3.5. For any morphismβ → α of I, let fαβ : Xβ → Xα be the corresponding morphism ofSpf(K+)-schemes. More-over, for everyα ∈ I, let Ωtf

Xα/K+ be the image of the morphismΩXα/K+ → ΩXα/K+ ⊗K+ K

(“ tf” stays for torsion-free). The following two conditions areequivalent :

(i) The system(Xα | α ∈ I) is deeply ramified.(ii) For everyα ∈ I there is a morphismβ → α of I, such that the image of the natural

morphismf∗αβ(ΩtfXα/K+)→ Ωtf

Xβ/K+ is contained in the subsheafa · ΩtfXβ/K+ .

Proof. It is clear that (ii)⇒(i). We show that (i)⇒(ii). Under the above assumptions, everyXα

is quasi-compact, hence we can cover it by finitely many affineformal schemesUi := Spf(Ai)(i = 1, ..., n) of finite type overSpf(K+). Then, for everyi = 1, ..., n, the restriction ofΩXα/K+

to Ui is the coherent sheaf(ΩanAi/K+) (notation of [18, Ch.I,§10.10.1]). HenceΩtf

Xα/K+ is acoherent sheaf ofOXα-modules. For every morphismβ → α, let

Uαβ := x ∈ Xβ | Im(f∗αβ(ΩtfXα/K+)x → (Ωtf

Xβ/K+)x) ⊂ a · (ΩtfXβ/K+)x.

Uαβ is therefore a constructible open subset ofXβ, and we denote its complement byZαβ. Byassumption (i) we know that

limβ→α

Zαβ =⋂

β→α

π−1α (Zαβ) = ∅.

If we retopologize the reduced schemesZαβ by their constructible topologies, we get an inversesystem of compact spaces, and deduce that someZαβ is empty by [13, Ch.I,§9, n.6, Prop.8(b)].

Example 6.3.7.The prototype of deeply ramified systems is given by the towerof morphisms

...→ BdK+(0, ρ1/pn

)φn→ BdK+(0, ρ1/p

n−1

)φn−1→ ...

φ1→ BdK+(0, ρ)(6.3.8)


where, for anyr = (r1, ..., rd) ∈ (K×)d, we have denoted

BdK+(0, |r|) := Spf(K+〈r−11 T1, ..., r−1d Td〉)

(i.e., the formald-dimensional polydisc defined by the equations|Ti| ≤ |ri|, i = 1, ..., d). Themorphismsφn are induced by the ring homomorphismsTi 7→ T pi (i = 1, ..., d). Notice that thetower (6.3.8) is defined wheneverρi ∈ Γp

∞

K :=⋂n∈N Γ

pn

K for everyi = 1, ..., d. We leave to thereader the verification that condition (ii) of lemma 6.3.6 isindeed satisfied.

Lemma 6.3.9. LetX := (Xα | α ∈ I) be a cofiltered system as in definition 6.3.5.(i) If Y := (Yα | α ∈ I) → (Xα | α ∈ I) is a morphism of cofiltered systems such that the

induced morphisms of adic spacesd(Yα) → d(Xα) are unramified for everyα ∈ I (cp.(6.2.6.2.20)), thenY is deeply ramified ifX is.

(ii) Let Z := (Zβ | β ∈ J) be another such cofiltered system, and suppose thatX andZ areisomorphic as pro-objects of the category of formal schemes. ThenX is deeply ramified ifand only ifZ is.

(iii) If X andZ := (Zβ | β ∈ J) are two deeply ramified cofiltered systems, then the fibredproductX× Z := (Xα ×Spf(K+) Zβ | (α, β) ∈ I × J) is deeply ramified.

Proof. (i): by [9, Prop.2.2] the natural morphismΩX∞/K+ ⊗K+ K → ΩY∞/K+ ⊗K+ K is anepimorphism; the claim follows easily. (ii) and (iii) are easy and shall be left to the reader.

6.3.6.3.10. Let nowX := (Xα | α ∈ I) be a deeply ramified cofiltered system, with transitionmorphismsfαβ : Xβ → Xα corresponding to morphismsβ → α of I. We define a new cofilteredsystemd(X) as follows. The indexing categoryd(I) consists of all the pairs(α,Y → Xα),whereα ∈ I andY → Xα is any morphism of formal schemes of finite presentation overSpf(K+) such that the induced morphism of adic spacesd(Y)→ d(Xα) is an isomorphism. Amorphism(β,Y→ Xβ)→ (α,Z→ Xα) is a commutative diagram

Y //

Xβ

fαβ

Z // Xα.

The system itself is the obvious functorF fromd(I) to the category of formal schemes such thatF (α,Y → Xα) = Y. Using lemma 6.3.9(i) one sees easily thatd(X) is still deeply ramified.Morever, taking (6.2.6.2.21) into account, we obtain a natural isomorphism of topologicallyringed spaces:

d(X∞) := limd(I)

F∼−→ lim

α∈I(d(Xα),O

+d(Xα)

).

For this reason, we shall denote byO+d(X∞) the structure sheaf ofd(X∞). For everyx ∈ d(X∞),

we letκ(x)+ := O+d(X∞),x/

⋂n∈N a

n · O+d(X∞),x. The ringκ(x)+ is a filtered colimit of valuation

rings, hence it is a valuation ring. Moreover, the image ofa in κ(x)+ is topologically nilpotentfor the valuation topology ofκ(x)+.

Proposition 6.3.11.Let X := (Xα | α ∈ I) be a deeply ramified cofiltered system. Then, forevery pointx ∈ d(X∞), the valuation ringκ(x)+ is deeply ramified.

Proof. For everyK+-moduleM , let us denote byTn(M) the submodule ofM annihilated byan. Furthermore, letT (M) :=

⋃n∈N Tn(M). Since the cofiltered systemd(X) is deeply ram-

ified, we have:(Ωd(X∞)/K+)x = T (Ωd(X∞)/K+)x + a · (Ωd(X∞)/K+)x. To lighten notation, letO+x := O

+d(X∞),x. From lemma 6.2.29 one deduces easily that the natural mapΩ

O+x /K+ →

(Ωd(X∞)/K+)x is surjective witha-divisible kernel. Hence, by snake lemma, the induced mapTn(ΩO

+x /K+)→ Tn(Ωd(X∞)/K+)x is surjective for everyn, anda fortiori the mapT (Ω

O+x /K+)→


T (Ωd(X∞)/K+)x is onto. It follows easily thatΩO

+x /K+ = T (Ω

O+x /K+) + a ·Ω

O+x /K+ , and conse-

quently:Ωκ(x)+/K+ = T (Ωκ(x)+/K+) + a · Ωκ(x)+/K+ . However, it follows easily from theorem5.6.11 thatT (Ωκ(x)+/K+) = 0, so finallyΩκ(x)+/K+ = a · Ωκ(x)+/K+ and

Lκ(x)+/K+

L

⊗K+ K+/a ·K+ ≃ 0.(6.3.12)

On the other hand, if(E, | · |E) is any valued field extension ofκ(x)+, we have

Hi(LE+/K+ ⊗K+ K+/a ·K+) = 0 for all i > 0(6.3.13)

by theorem 5.6.11. From (6.3.12), (6.3.13) and transitivity for the towerK+ ⊂ κ(x)+ ⊂ E+,we deriveHi(LE+/κ(x)+ ⊗K+ K+/a · K+) = 0 for all i > 0.. Again by theorem 5.6.11 andremark 5.6.14 we conclude.

6.3.6.3.14. LetX be a cofiltered system as in definition 6.3.5. LetA be a sheaf ofO+d(X∞)-

algebras. We say thatA is a weaklyetaleO+d(X∞)-algebra if, for everyx ∈ d(X∞), the stalk

A ax is a weakly etaleO+a

d(X∞),x-algebra.

Theorem 6.3.15.Suppose thatK is deeply ramified, and letX := (Xα | α ∈ I) be a deeplyramified cofiltered system. Let alsof : Y := (Yα | α ∈ I) → X be a morphism of cofilteredsystems, such that the induced morphismsd(Yα) → d(Xα) are finiteetale for everyα ∈ I.Thenf∞∗O

+ad(Y∞) is a weaklyetaleO

+ad(X∞) algebra.

Proof. To lighten notation, let us writeO+ (resp. A +) instead ofO+d(X∞) (resp. f∞∗O

+d(Y∞)).

For everyα ∈ I consider the cofiltered systemZ(α) := X×XαYα indexed byI/α (the categoryof morphismsβ → α), which is defined by settingZ(α)β→α := Xβ ×Xα Yα. We have obviousmorphisms of cofiltered systemsf/α : Z(α) → X; the sheafA + is the colimit of the sheavesf/α∞∗O

+d(Z(α)∞). Hence it suffices to prove the assertion for the latter sheaves, and therefore in

order to show the theorem we can and do assume that there exists α ∈ I such that, for everyβ → α, the induced commutative diagram

d(Yβ) //

d(Yα)

d(Xβ) // d(Yα)

is cartesian. Letx ∈ d(X∞).

Claim 6.3.16. A +x is a flatO+

x -algebra.

Proof of the claim:On one hand, by assumptionA +x [1/a] is a flatO+

x [1/a]-algebra; on theother hand,A +

x ⊗O+xκ(x)+ is a flatκ(x)+-module by claim 6.3.3, so thatA +

x /a ·A+x is a flat

O+x /a · O

+x -module; thus the claim follows from [31, Partie II, lemme 1.4.2.1].

Let e ∈ C := A +x ⊗O

+x

A +x [1/a] be the idempotent provided by lemma 3.1.4. In view of

claim 6.3.16, we only have to show thatε · e ∈ C+ := A +x ⊗O

+x

A +x for everyε ∈ m. Let e be

the image ofe in C ⊗Ox κ(x). SetI :=⋂n≥0 a

n · O+x .

Claim 6.3.17. (A +x /IA

+x )a is an etale(κ(x)+)a-algebra.

Proof of the claim:Let y be the maximal generization ofx in d(X∞); thenκ(y)+ is a rank onevaluation and the specialisation map induces isomorphismsκ(x)+a

∼→ κ(y)+a, (A +

x /IA+x )a

∼→

(A +y /IA

+y )a (recall that the standing basic setup is the standard setup of K+). Hence, we can

replacex by y, and assume thatκ(x)+ has rank one. In view of propositions 2.4.14, 6.3.11 and5.6.2 it suffices to show thatA +

x /IA+x is the integral closure ofκ(x)+ in Ax/IAx. To this

aim, we remark first of all that the natural mapA +x /IA

+x → Ax/IAx is injective : indeed,


sinceI = a · I, we haveIAx = IA +x . Next, we remark also thatA +

x is the integral closure ofO+x in Ax; indeed, this follows from [7,§6.2.2, Lemma 3, Prop.2], after taking colimits. This

already shows thatA +x /IA

+x is a subalgebra ofAx/IAx integral overκ(x)+. To conclude,

suppose thatf ∈ Ax/IAx satisfies an integral equation:fn+b1 ·f

n−1+ ...+bn = 0, for certain

b1, ..., bn ∈ A +x /IA

+x ; pick arbitrary representativesf ∈ Ax, bi ∈ A +

x of these elements. Itfollows thatfn + b1 · fn−1 + ... + bn ∈ IAx. SinceIAx = IA +

x , it follows thatf is integraloverA +

x , sof ∈ A +x and the claim follows.

Henceε · e ∈ C+ ⊗O

+xκ(x)+ for everyε ∈ m. Let eε ∈ C+ be any lifting ofε · e; then

ε · e − eε is in the kernel of the projectionC → C ⊗O

+xκ(x). Let n ∈ N be a large enough

integer, so thatan · e ∈ C+; it follows thatan · (ε · e − eε) is in the kernel of the projectionC+ → C+ ⊗

O+xκ(x)+, consequentlyan · (ε · e − eε) = an · c for somec ∈ C+. Sincea is a

non-zero-divisor inC+, it follows thatε · e = c+ eε ∈ C+, as required.

6.4. Semicontinuity of the discriminant.

Definition 6.4.1. Let (V,m) be a basic setup,A a V a-algebra andP an almost projectiveA-module of constant rankr ∈ N. Suppose moreover thatP is endowed with a bilinear formb : P ⊗A P → A. We let β : P → P ∗ be theA-linear morphism defined by the rule:β(x)(y) := b(x⊗ y) for everyx, y ∈ P∗. Thediscriminantof the pair(P, b) is the ideal

dA(P, b) := AnnA(Coker(ΛrAβ : ΛrAP → ΛrAP

∗)).

6.4.6.4.2. As a special case, we can consider the pair(B, tB/A) consisting of anA-algebraBwhich is almost projective of constant rankr overA, and its trace formtB/A. In this situation,we letdB/A := dA(B, tB/A), and we call this ideal thediscriminant of theA-algebraB.

Lemma 6.4.3. LetB be an almost projectiveA-algebra of constant rankr as anA-module.ThenB is etale overA if and only if dB/A = A.

Proof. By theorem 4.1.14, it is clear thatdB/A = A whenB is etale overA. Suppose thereforethat dB/A = A; it follows thatΛrAτB/A is an epimorphism. However, by proposition 4.3.27,ΛrAB andΛrAB

∗ are invertibleA-modules. It then follows by lemma 4.1.5(iv) thatΛrAτB/A is anisomorphism, henceφ is an isomorphism, by virtue of proposition 4.4.28.

Lemma 6.4.4. Let (V,m) be the standard setup associated to a valued field(K, | · |) (cp.(5.1.5.1.15), especially,V := K+). Let P ′ ⊂ P be two almost projectiveV a-modules ofconstant rank equal tor. Let b : P ⊗V a P → V a be a bilinear form, such thatb ⊗V a 1Ka is aperfect pairing, and denote byb′ the restriction ofb toP ′ ⊗V a P ′. Then we have:

dV a(P ′, b′) = F0(P/P′)2 · dV a(P, b).

Proof. Let j : P ′ → P be the imbedding,β : P → P ∗ (resp. β ′ : P ′ → P ′∗) theV a-linearmorphism associated tob (resp. tob′). The assumptions implies thatΛrV aj, ΛrV aj∗, ΛrV aβ andΛrV aβ ′ are all injective, and clearly we have

ΛrV aj∗ ΛrV aβ ΛrV aj = ΛrV aβ ′.

There follow short exact sequences:

0→ Coker(ΛrV aβ)→ Coker(ΛrV a(j∗ β))→ Coker(ΛrV aj∗)→ 00→ Coker(ΛrV aj)→ Coker(ΛrV aβ ′)→ Coker(ΛrV a(j∗ β))→ 0.

Using lemma 5.3.1 and remark 5.3.5(ii), we deduce:

F0(Coker(ΛrV aβ ′)) = F0(Coker(Λ

rV aj)) · F0(Coker(Λ

rV aβ)) · F0(Coker(Λ

rV aj∗)).

However,ΛrV aP and ΛrV aP ′ are invertibleV a-module by proposition 4.3.27, consequentlyF0(CokerΛ

rV aβ ′)) = dV a(P ′, b′), F0(Coker(Λ

rV aβ)) = dV a(P, b) andF0(Coker(Λ

rV aj)) =

F0(Coker(ΛrV aj∗)) = F0(P/P

′).


6.4.6.4.5. After these generalities, we return to the standard setup(K+,m) of this chapter,associated to a valued field(K, | · |) of rank one (cp. (5.1.5.1.15)). Consider an etaleK-algebraL; we denote byWL the integral closure ofK+ in L. L is the product of finitely many separablefield extensions ofK, thereforeW a

L is an almost projectiveK+a-module of constant rankn, byproposition 5.3.6. Hence, the discriminant ofW a

L overK+a is defined, and to lighten notation,we will denote it byd+L/K . Furthermore, sinceL is etale overK, it is clear thatd+L/K is afractional ideal ofK+a (cp. (5.1.5.1.16)). Let| · | : Div(K+a) → Γ∧K be the isomorphismprovided by lemma 5.1.19. We obtain an element|d+L/K | ∈ Γ∧K ; after choosing (cp. example(5.1.21)(vi)) an order preserving isomorphism

((ΓK ⊗Z Q)∧,≤)∼→ (R>0,≤)(6.4.6)

on the multiplicative group of positive real numbers, we canthen view|d+L/K | ∈ (0, 1].

Lemma 6.4.7. Let K, L be as in(6.4.6.4.5)and denote byK∧ the completion ofK for thevaluation topology. SetL∧ := K∧ ⊗K L. ThenK∧+ ⊗K+ d+L/K = d+L∧/K∧.

Proof. Since the base changeK → K∧ is faithfully flat, everything is clear from the definitions,once we have established thatWL∧ ≃ K∧+ ⊗K+ WL. However, both rings can be identifiedwith thea-adic completion(WL)

∧ of WL, so the assertion follows.

6.4.6.4.8. LetX be an adic space locally of finite type overSpa(K,K+). X is a locallyspectral space, and every pointx ∈ X admits a unique maximal generisationr(x) ∈ X. Thevaluation ringκ(r(x))+ has rank one, and admits a natural imbeddingK+ ⊂ κ(r(x))+, contin-uous for the valuation topologies; especially, the image ofthe topologically nilpotent elementais topologically nilpotent inκ(r(x))+. This imbedding induces therefore a natural isomorphismof completed value groups

(ΓK ⊗Z Q)∧∼→ (Γκ(r(x)) ⊗Z Q)∧.

In particular, our original choice of isomorphism (6.4.6) fixes univocally a similar isomorphismfor every pointr(x). We denote byM(X) the setr(X) endowed with the quotient topologyinduced by the mappingX → r(X) : x 7→ r(x). This topology is coarser than the subspacetopology induced by the imbedding intoX. The mappingx 7→ r(x) is a continuous retractionof X onto the subsetM(X) of its maximal points. IfX is a quasi-separated quasi-compactadic space,M(X) is a compact Hausdorff topological space.

6.4.6.4.9. LetX be as in (6.4.6.4.8), and letf : Y → X be a finite etale morphism of adicspaces. For every pointx ∈ X, the fibreE(x) := (f∗OY )x ⊗OX,x

κ(x) is a finite etaleκ(x)-algebra. If nowx ∈M(X), we can consider the discriminantd+E(x)/κ(x) defined as in (6.4.6.4.5)(warning: notice that the definition makes sense when we choose the standard setup associatedto the valuation ringκ(x)+; since it may happen that the valuation ofK is discrete and thatof κ(x) is not discrete, the setups relative toK and toκ(x) may not agree in general). Uponpassing to absolute values, we finally obtain a real valued function:

d+Y/X :M(X)→ (0, 1] x 7→ |d+E(x)/κ(x)|.

The study of the functiond+Y/X is reduced easily to the case whereX (henceY ) are affinoid. Insuch case, one can state the main result in a more general form, as follows.

Definition 6.4.10. LetA be any (commutative unitary) ring.

(i) We denote byN (A) the set consisting of all multiplicative seminorms| · | : A → R.For everyx ∈ N (A) and f ∈ A we write usually|f(x)| in place ofx(f). N (A) isendowed with the coarsest topology such that, for everyf ∈ A, the real-valued map|f | : N (A)→ R given by the rule:x 7→ |f(x)|, is continuous.


(ii) For everyx ∈ N (A), we letSupp(x) := f ∈ A | |f(x)| = 0. ThenSupp(x) is a primeideal and we setκ(x) := Frac(A/Supp(x)). The seminormx induces a valuation on theresidue fieldκ(x), and as usual we denote byκ(x)+ its valuation ring.

(iii) Let A→ B be a finite etale morphism. For everyx ∈ N (A), we letE(x) := B ⊗A κ(x).ThenE(x) is an etaleκ(x)-algebra, so we can defined+B/A(x) := d+E(x)/κ(x) (cp. thewarning in (6.4.6.4.9)). By settingx 7→ |d+B/A(x)| we obtain a well-defined function

|d+B/A| : N (A)→ (0, 1].

6.4.6.4.11. IfX = Spa(A,A+), with A a completeK-algebra of topologically finite type,thenM(X) is naturally homeomorphic to the subspaceM(A) of N (A) consisting of the con-tinuous seminorms that extend the absolute value ofK given by (6.4.6). It is shown in [5,§1.2]thatM(A) is a compact Hausdorff space, for every BanachK-algebraA.

Proposition 6.4.12.Let A be a ring,B a finite etaleA-algebra. Then the function|d+B/A| islower semi-continuous (i.e. it is continuous for the topology of(0, 1] whose open subsets are ofthe form(c, 1], c ∈ [0, 1]).

Proof. Let f ∈ A be any element; notice thatN (A[1/f ]) is naturally homeomorphic to theopen subsetU(f) := x ∈ N (A) | |f(x)| 6= 0. Hence, after replacingA by some localization,we can assume thatB is a freeA-module, say of rankn. For everyb ∈ B, let χ(b, T ) :=T n + s1(b) · T n−1 + ...+ sn(b) be the characteristic polynomial of theA-linear endomorphismB → B given by the ruleb′ 7→ b′ · b.

Claim6.4.13. For every pointx ∈ N (A) and everyb ∈ B, the following are equivalent:

(i) b⊗ 1 ∈ WE(x).(ii) |si(b)(x)| ≤ 1 for i = 1, ..., n.

Proof of the claim:Indeed, if (ii) holds, then the image ofχ(b, T ) in κ(x)[T ] is a monic poly-nomial with coefficients inκ(x)+ andb ⊗ 1 is one of its roots (Cayley-Hamilton), henceb ⊗ 1

is integral overκ(x)+, which is (i). Conversely, if (i) holds, letE(x) =∏k

j=1Ej be the de-composition ofE(x) as product of finite separable extensions ofκ(x), and letbj ∈ Ej be theimage ofb. It follows thatbj ∈ WEj

for everyj = 1, ..., k, and moreover the imageχ(b, T )of χ(b, T ) in κ(x)[T ] decomposes as a product

∏kj=1 χ(bj , T ). It suffices therefore to show

that the coefficientssij(bj) satisfy (ii) for everyi ≤ n and j ≤ k, so we can assume thatE(x) is a field. Letmb(T ) ∈ κ(x)[T ] be the minimal polynomial ofb ⊗ 1; it is well knownthatχ(b, T ) dividesmb(T )

n, hence the roots ofχ(b, T ) are conjugates ofb under the action ofG := Gal(κ(x)a/κ(x)). LetC be the integral closure ofκ(x)+ in a finite Galois extension ofκ(x) containingE(x); C is an integralκ(x)+-algebra and the Galois conjugates ofb⊗ 1 are allcontained inC. Since the latter are the roots ofχ(b, T ), the elementssi(b) ⊗ 1 are symmetricpolynomials of the elementsσ(b⊗ 1) (σ ∈ G), sosi(b)⊗ 1 ∈ C ∩κ(x) = κ(x)+, which is (ii).

Let tB/A be the trace morphism of theA-algebraB, and letx ∈ N (A). Then the tracemorphismtx := tE(x)/κ(x) equalstB/A ⊗A 1κ(x).

Claim6.4.14. For every real numberε > 0 we can find a freeκ(x)+-submoduleWε of WE(x),such thatWε ⊗κ(x)+ κ(x) = E(x) and

|dκ(x)+(Wε, tx)|+ ε > |d+B/A(x)|.(6.4.15)

Proof of the claim:From claim 5.3.8, we derive that, for every positive real numberδ < 1 thereexists a free finitely generatedκ(x)+-submoduleWδ ⊂ WE(x) such that|F0(W

aE(x)/W

aδ )| > δ;

in view of lemma 6.4.4, the claim follows easily.


Let w1, ..., wn be a basis ofWε; up to replacingA by a localizationA[1/g] for someg ∈ A,we can writewi = bi ⊗ 1, for somebi ∈ B (i = 1, ..., n). Consequently:

|dκ(x)+(Wε, tx)| = | det(tB/A(bi ⊗ bj))(x)|.(6.4.16)

By claim 6.4.13 we have|sj(bi)(x)| ≤ 1 for i, j = 1, ..., n. Let 1 > δ > 0 be a real number; foreveryi, j ≤ n, we define an open neighborhoodUij of x inN (A) as follows. Suppose first that|sj(bi)(x)| < 1; since the real-valued functiony 7→ |sj(bi)(y)| is continuous onN (A) (for thestandard topology ofR), we can findUij such that|sj(bi)(y)| ≤ 1 for all y ∈ Uij .

Suppose next that|sj(bi)(x)| = 1; then, up to replacingA byA[1/sj(bi)], we can assume thatsj(bi) is invertible inA. We pickUij such that

|sj(bi)(y)| ≤ 1 + δ for everyy ∈ Uij.(6.4.17)

We setU :=⋂

1≤i,j≤n Uij . Next, we define, for everyy ∈ U , an elementcy ∈ A, as follows.Chooseα, β ≤ n such that|sβ(bα)(y)| = max1≤i,j≤n |sj(bi)(y)|. If |sβ(bα)(y)| ≤ 1, then setcy := 1; if |sβ(bα)(y)| > 1, setcy := sβ(bα)

−1. Then|sj(cy · bi)(y)| ≤ 1 for everyi, j = 1, ..., nand everyy ∈ U . LetW y be theκ(y)+-submodule ofB ⊗A κ(y)+ spanned by the images ofcy · b1, ..., cy · bn. It follows thatW y ⊂WE(y) for everyy ∈ U . We compute:

|d+Y/X(y)| ≥ |dκ(y)+(W y, ty)|= | det(tB/A(cy · bi ⊗ c(y) · bj))(y)|

= |cy(y)|2n · | det(tB/A(bi ⊗ bj))(y)|.

However, the real-valued functiony 7→ | det(tB/A(bi ⊗ bj))(y)| is continuous onN (A), there-fore, combining (6.4.15) and (6.4.16), we see that, up to shrinking further the open neighbor-hoodU , we can assume that| det(tB/A(bi ⊗ bj))(y)|+ ε > |d+B/A(x)| for all y ∈ U , so finally:

|d+B/A(y)| ≥ (1− δ)2n · (|d+B/A(x)| − ε) for everyy ∈ U

which implies the claim.

Theorem 6.4.18.LetY → X be as in(6.4.6.4.9). Then the mapd+Y/X is lower semi-continuous.

Proof. We can assume thatX = Spa(A,A+), whereA is a completeK-algebra of topologicallyfinite type, and thereforeY = Spa(B,B+), for a finite etaleA-algebraB. Then there is a naturalhomeomorphismω :M(X)

∼→M(A), so the theorem follows from proposition 6.4.12 and :

Claim 6.4.19. d+Y/X = d+B/A ω.

Proof of the claim:Let x ∈ M(X); x corresponds to a rank one valuation ofA, whose valuegroup we identify with (a subgroup of)R according to (6.4.6.4.8). The resulting multiplicativeseminorm isω(x). We derive easily a natural imbeddingι : κ(ω(x)) ⊂ κ(x), compatible withthe identifications of value groups. One knows moreover thatι induces an isomorphism oncompletionsι∧ : κ(ω(x))∧

∼→ κ(x)∧, so the claim follows from lemma 6.4.7.

Lemma 6.4.20.Let(Kα, |·|α | α ∈ I) be a system of valued field extensions of(K, |·|), indexedby a filtered small categoryI and such thatK+

α is a valuation ring of rank one for everyα ∈ I.Let moreoverL be a finiteetaleKβ-algebra, for someβ ∈ I. SetLα := L ⊗Kβ

Kα for everymorphismβ → α in I. Then :

(i) (K∞, | · |∞) := colimα∈I

(Kα, | · |α) is a valued field extension of(K, | · |), with valuation ring

of rank one.(ii) SetL∞ := L⊗Kβ

K∞. Then, for every sequence of morphismsβ → γ → α in I, we have|d+Lα/Kα

| ≥ |d+Lγ/Kγ|, and moreover: lim

(β→α)∈I|d+Lα/Kα

| = |d+L∞/K∞|.


Proof. (i) is obvious. The proof of the first assertion in (ii) is easyand shall be left to the reader.For the second assertion in (ii) we remark that, due to claim 5.3.8, for everyε < 1 there existsa freeK+

∞-submoduleWε ⊂ WL∞ of finite type, such that|F0(WaL∞/W a

ε )| > ε. We can findα ∈ I such thatWε = W0 ⊗K+

αK+∞ for some freeK+

α submoduleW0 ⊂ L+α . It then follows

from lemma 6.4.4 that

|d+Lα/Kα| ≥ |dK+

α(W0, tLα/Kα)| = |dK+

∞(Wε, tL∞/K∞)| > ε2 · |d+L∞/K∞

|

for every morphismα→ β in I.

6.4.6.4.21. Suppose now that(K, | · |) is deeply ramified and letX := (Xα | α ∈ I) be a deeplyramified cofiltered system of formal schemes of finite type over Spf(K+). Suppose furthermorethat it is given, for someβ ∈ I, a morphismYβ → Xβ of formal Spf(K+)-schemes of finitetype, such that the induced morphismd(Yβ) → d(Xβ) is finite and etale. For every morphismα → β of I we setYα := Yβ ×Xβ

Xα and denote byfα : d(Yα) → d(Xα) the inducedmorphism of adic spaces.

Theorem 6.4.22.In the situation of(6.4.6.4.21), for every positive real numberε < 1 thereexists a morphismα→ β in I such that, for every morphismγ → α we have

|d+d(Yγ )/d(Xγ )(x)| > ε for everyx ∈M(d(Xγ)).

Proof. Notice thatY := (Yα | α → β) is a cofiltered system, hence we can defined(X∞) andd(Y∞) as in (6.3.6.3.10), and we obtain a morphism of locally ringed spacesf∞ : d(Y∞) →d(X∞). For everyα ∈ I, let πα : d(X∞) → d(Xα) be the natural morphism. Moreover, letM(d(X))∞ := lim

α∈IM(d(Xα)); as a topological space, it is compact, due to (6.2.6.2.20);as

a set, it admits an injective (usually non-continuous) mapM(d(X))∞ → d(X∞), so we canidentify it as a subset of the latter.

Let x ∈ M(d(X))∞; by proposition 6.3.11, the valuation ringκ(x)+ is deeply ramified. Setκ(x) := κ(x)+ ⊗K+ K; it is clear that the morphism

κ(x)→ E(x) := (f∞∗O+d(Y∞))x ⊗κ(x)+ κ(x)

is finite and etale. LetWx be the integral closure ofκ(x)+ in E(x). By proposition 5.6.2 wededuce easily that the induced morphism ofK+a-algebras(κ(x)+)a → (Wx)

a is weakly etale,hence etale by proposition 5.3.6. Consequently|d+E(x)/κ(x)| = 1, in light of lemma 6.4.3. Foreveryα→ β, letxα := πα(x). Thenκ(x) is the colimit of the filtered system(κ(xα) | α→ β),and similarlyE(x) is the colimit of the finite etaleκ(xα)-algebras(fβ∗Od(Yβ))xβ⊗O

+d(Xβ ),xβ

κ(xα)

(for all α → β). In this situation, lemma 6.4.20 applies and shows that, for everyε < 1 thereexistsα(ε, x) such that

|d+d(Yα)/d(Xα)(xα)| > ε for everyα→ α(ε, x).(6.4.23)

In light of theorem 6.4.18, for everyα→ β, the subset

Xα(ε) := y ∈M(d(Xα)) | |d+d(Yα)/d(Xα)

(y)| ≤ ε

is closed inM(d(Xα)), hence compact. From (6.4.23) we see thatlimα→β

Xα(ε) = ∅, therefore

one of theXα(ε) must be empty ([13, Ch.I,§9, n.6, Prop.8(b)]), and the claim follows.

6.4.6.4.24. Let us choose an imbeddingρ : (ΓK , ·,≤) → (R,+,≤) as in example 5.1.21(vi).Let f : Y → X be a finite etale morphism of adic spaces of finite type overSpa(K,K+). Foreveryx ∈ X, setA(x) := (f∗O

+Y )x⊗O

+X,x

(f∗O+Y )x; we denote byex ∈ A(x)⊗K+K the unique


idempotent characterized by the conditions of proposition3.1.4. We define thedefectof themorphismf as the real number

def(f) := infr ∈ R | ε · ex ∈ A(x) for everyx ∈ X and everyε ∈ K+ with ρ(|ε|) ≤ −r .

Clearlydef(f) ≥ 0 anddef(f) = 0 if and only if (f∗O+Y )

ax is an etaleO+a

X,x-algebra for everyx ∈ X. Furthermore we remark that, by proposition 2.4.14, the mapO

+aX,x → (f∗O

+Y )

ax is

weakly etale if and only if it is etale.

Corollary 6.4.25. In the situation of(6.4.6.4.21), for every real numberr > 0 there existsα ∈ I such that, for every morphismγ → α, we have:def(fγ) < r.

Proof. Let r > 0; according to theorem 6.4.22, there existsα ∈ I such thatε · ex ∈ A(x) foreveryγ → α, everyx ∈ M(d(Xγ)) and everyε ∈ K+ with ρ(|ε|) ≤ −r. If now y ∈ d(Xγ)is any point, there is a unique generisationx of y inM(d(Xγ)). Let φ : A(y) → A(x) be theinduced specialisation map, and setφK := φ⊗K+ 1K . One verifies easily that

φK(A(y)⊗K+ K) ∩m · A(x) ⊂ φ(A(y)).

SinceφK(ey) = ex, the claim follows easily.

6.4.6.4.26. To conclude, we want to explain briefly what kindof Galois cohomology calcula-tions are enabled by the results of this section. Letf : Y → X be a finite etaleGaloismorphismof Spa(K,K+)-adic spaces of finite type, and letG denote the group ofX-automorphisms ofY . Denote byf∗O

+Y [G]-Mod the category off∗O

+Y -modules onX, endowed with a semilin-

ear action ofG. Let ΓG : f∗O+Y [G]-Mod → O

+X-Mod be the functor that associates to an

f∗O+Y [G]-module the sheaf of itsG-invariant local sections. A standard argument shows that,

for everyf∗O+Y [G]-modulesF , the cone of the natural morphism inD(f∗O

+Y -Mod)

O+Y ⊗O

+XRΓGF → F(6.4.27)

is annihilated by allε ∈ m such thatρ(ε) < −def(f). However, for applications one is rathermore interested in understanding the Galois cohomology groupsH i := H i(G,H0(X,F )).One can try to studyH i via (6.4.27); indeed, a bridge between these two objects is providedby the higher derived functors of the related functorΓG : f∗O

+Y [G]-Mod → Γ(X,O+)-Mod,

defined byF 7→ Γ(X,ΓGF ) = Γ(X,F )G. We have two spectral sequences converging toRΓGF , namely

Epq2 : Hp(X,RqΓGF )⇒Rp+qΓGF

F pq2 : Hp(G,Hq(X,F ))⇒Rp+qΓGF .

Using (6.4.27) one deduces thatEpq2 degenerates up to some torsion, which can be estimated

precisely in terms of the defect of the morphismf . However, the spectral sequenceF pq2 contains

the termsHq(X,F ), about which not much is currently known. In this direction,the onlyresults that we could found in the literature concern the calculation ofH i(Y,O+

Y ), for an affinoidspace, under some very restrictive assumptions : in [3] these groups are shown to be almostzero modules fori > 0, in caseY admits a smooth formal model overK+; in [49] the case ofgeneralized polydiscs is taken up, and the same kind of almost vanishing is proven.


7. APPENDIX

7.1. In this appendix we have gathered a few miscellaneous results that were found in thecourse of our investigation, and which may be useful for other applications.

7.1.7.1.1. We need some preliminaries on simplicial objects : first of all, asimplicial almostalgebrais just an object in the categorys.(V a-Alg). Then for a given simplicial almost algebraA we have the categoryA-Mod of A-modules : it consists of all simplicial almostV -modulesM such thatM [n] is anA[n]-module and such that the face and degeneracy morphismsdi :M [n]→M [n− 1] andsi :M [n]→M [n + 1] (i = 0, 1, ..., n) areA[n]-linear.

7.1.7.1.2. We will need also the derived category ofA-modules; it is defined as follows. Abit more generally, letC be any abelian category. For an objectX of s.C let N(X) be thenormalized chain complex (defined as in [34, I.1.3]). By the theorem of Dold-Kan ([50, Th.8.4.1])X 7→ N(X) induces an equivalenceN : s.C → C•(C ) with the categoryC•(C ) ofchain complexes of object ofC that vanish in positive degrees. Now we say that a morphismX → Y in s.C is aquasi-isomorphismif the induced morphismN(X) → N(Y ) is a quasi-isomorphism of chain complexes.

7.1.7.1.3. In the following we fix a simplicial almost algebraA.

Definition 7.1.4. We say thatA is exactif the almost algebrasA[n] are exact for alln ∈ N.A morphismφ : M → N of A-modules (orA-algebras) is aquasi-isomorphismif the mor-phismφ of underlying simplicial almostV -modules is a quasi-isomorphism. We define thecategoryD•(A) (resp. the categoryD•(A-Alg)) as the localization of the categoryA-Mod

(resp.A-Alg) with respect to the class of quasi-isomorphisms.

7.1.7.1.5. As usual, the morphisms inD•(A) can be computed via a calculus of fraction onthe categoryHot•(A) of simplicial complexes up to homotopy. Moreover, ifA1 andA2 aretwo simplicial almost algebras, then the “extension of scalars” functors define equivalences ofcategories

D•(A1 × A2)∼−→ D•(A1)× D•(A2)

D•(A1 × A2-Alg)∼−→ D•(A1-Alg)× D•(A2-Alg).

Proposition 7.1.6. LetA be a simplicialV a-algebra.

(i) The functor onA-algebras given byB 7→ (s.V a×B)!! preserves quasi-isomorphisms andtherefore induces a functorD•(A-Alg)→ D•((s.V a × A)!!-Alg).

(ii) The localisation functorR 7→ Ra followed by “extension of scalars” vias.V a × A → Ainduces a functorD•((s.V a × A)!!-Alg) → D•(A-Alg) and the composition of this andthe above functor is naturally isomorphic to the identity functor onD•(A-Alg).

Proof. (i) : let B → C be a quasi-isomorphism ofA-algebras. Clearly the induced morphisms.V a × B → s.V a × C is still a quasi-isomorphism ofV -algebras. But by remark 2.2.26,s.V a × B and s.V a × C are exact simplicial almostV -algebras; moreover, it follows fromcorollary 2.2.22 that(s.V a × B)! → (s.V a × C)! is a quasi-isomorphism ofV -modules. Thenthe claim follows easily from the exactness of the sequence (2.2.24). Now (ii) is clear.

Remark 7.1.7. In casem is flat, then allA-algebras are exact, and the same argument showsthat the functorB 7→ B!! induces a functorD•(A-Alg)→ D•(A!!-Alg). In this case, composi-tion with localisation is naturally isomorphic to the identity functor onD•(A-Alg).

Proposition 7.1.8. Let f : R → S be a map ofV -algebras such thatfa : Ra → Sa is anisomorphism. ThenLaS/R ≃ 0 in D•(s.Sa).


Proof. We show by induction onq that

VAN (q;S/R) Hq(LaS/R) = 0.

For q = 0 the claim follows immediately from [34, II.1.2.4.2]. Therefore suppose thatq > 0and thatVAN (j;D/C) is known for all almost isomorphisms ofV -algebrasC → D and allj < q. LetR := f(R). Then by transitivity ([34, II.2.1.2]) we have a distinguished triangle inD•(s.Sa)

(S ⊗R LR/R)a u→ LaS/R

v→ La

S/R→ σ(S ⊗R LR/R)

a.

We deduce thatVAN (q;R/R) andVAN (q;S/R) imply VAN (q;S/R), thus we can assume thatf is either injective or surjective. LetS• → S be the simplicialV -algebra augmented overSdefined byS• := PV (S). It is a simplicial resolution ofS by freeV -algebras, in particularthe augmentation is a quasi-isomorphism of simplicialV -algebras. SetR• := S• ×S R. Thisis a simplicialV -algebra augmented overR via a quasi-isomorphism. Moreover, the inducedmorphismsR[n]a → S[n]a are isomorphisms. By [34, II.1.2.6.2] there is a quasi-isomorphismLS/R ≃ L∆

S•/R•. On the other hand we have a spectral sequence

E1ij := Hj(LS[i]/R[i])⇒ Hi+j(L

∆S•/R•

).

It follows easily thatVAN(j;S[i]/R[i]) for all i ≥ 0, j ≤ q impliesVAN(q;S/R). Thereforewe are reduced to the case whereS is a freeV -algebra andf is either injective or surjective.We examine separately these two cases. Iff : R→ V [T ] is surjective, then we can find a rightinverses : V [T ] → R for f . By applying transitivity to the sequenceV [T ] → R → V [T ] weget a distinguished triangle

(V [T ]⊗R LR/V [T ])a u→ LaV [T ]/V [T ]

v→ LaV [T ]/R → σ(V [T ]⊗R LR/V [T ])

a.

SinceLaV [T ]/V [T ] ≃ 0 there follows an isomorphism :Hq(LV [T ]/R)a ≃ Hq−1(V [T ]⊗RLR/V [T ])

a.Furthermore, sincefa is an isomorphism,sa is an isomorphism as well, hence by induction(and by a spectral sequence of the type [34, I.3.3.3.2])Hq−1(V [T ]⊗RLR/V [T ])

a ≃ 0. The claimfollows in this case.

Finally suppose thatf : R→ V [T ] is injective. WriteV [T ] = Sym(F ), for a freeV -moduleF and setF = m⊗V F ; sincefa is an isomorphism,Im(Sym(F )→ Sym(F )) ⊂ R. We applytransitivity to the sequenceSym(F ) → R → Sym(F ). By arguing as above we are reduced toshowing thatLa

Sym(F )/Sym(F )≃ 0.We know thatH0(LaSym(F )/Sym(F )

) ≃ 0 and we will show that

Hq(LaSym(F )/Sym(F )) ≃ 0 for q > 0. To this purpose we apply transitivity to the sequenceV →

Sym(F )→ Sym(F ). AsF andF are flatV -modules, [34, II.1.2.4.4] yieldsHq(LSym(F )/V ) ≃

Hq(LSym(F )/V ) ≃ 0 for q > 0 andH0(LSym(F )/V ) is a flatSym(F )-module. In particularHj(Sym(F ) ⊗Sym(F ) LSym(F )/V ) ≃ 0 for all j > 0. ConsequentlyHj+1(LSym(F )/Sym(F )) ≃ 0

for all j > 0 andH1(LSym(F )/Sym(F )) ≃ Ker(Sym(F ) ⊗Sym(F ) ΩSym(F )/V → ΩSym(F )/V ). Thelatter module is easily seen to be almost zero.

Theorem 7.1.9.Let φ : R → S be a map of simplicialV -algebras inducing an isomorphismRa ∼→ Sa in D•(Ra). Then(L∆

S/R)a ≃ 0 in D•(Sa).

Proof. Apply the base change theorem [34, II.2.2.1] to the (flat) projections ofs.V ×R ontoRand respectivelys.V to deduce that the natural mapL∆

s.V×S/s.V×R → L∆S/R⊕L

∆s.V/s.V → L∆

S/R isa quasi-isomorphism inD•(s.V ×S). By proposition 7.1.6 the induced morphism(s.V ×R)a!! →(s.V × S)a!! is still a quasi-isomorphism. There are spectral sequences

E1ij := Hj(L(V ×R[i])/(V×R[i])a!!

)⇒ Hi+j(L∆(s.V×R)/(s.V×R)a!!

)

F 1ij := Hj(L(V ×S[i])/(V×S[i])a!!

)⇒ Hi+j(L∆(s.V×S)/(s.V×S)a!!

).


On the other hand, by proposition 7.1.8 we haveLa(V×R[i])/(V ×R[i])a!! ≃ 0 ≃ La(V ×S[i])/(V×S[i])a!!for all i ∈ N. Then the theorem follows directly from [34, II.1.2.6.2(b)] and transitivity.

Proposition 7.1.10.LetA → B be a morphism of exact almostV -algebras. Then the naturalmapm⊗V LB!!/A!!

→ LB!!/A!!is a quasi-isomorphism.

Proof. By transitivity we may assumeA = V a. LetP• := PV (B!!) be the standard resolution ofB!! (see [34, II.1.2.1]). EachP [n]a containsV as a direct summand, hence it is exact, so that wehave an exact sequence of simplicialV -modules0→ s.m→ s.V ⊕ (P a

• )! → (P a• )!! → 0. The

augmentation(P a• )! → (Ba

!!)! ≃ B! is a quasi-isomorphism and we deduce that(P a• )!! → B!!

is a quasi-isomorphism; hence(P a• )!! → P• is a quasi-isomorphism as well. We haveP [n] ≃

Sym(Fn) for a freeV -moduleFn and the map(P [n]a)!! → P [n] is identified withSym(m ⊗VFn) → Sym(Fn), whenceΩP [n]a!!/V

⊗P [n]a!!P [n] → ΩP [n]/V is identified withm ⊗V ΩP [n]/V →

ΩP [n]/V . By [34, II.1.2.6.2] the mapL∆(P a

• )!!/V→ L∆

P•/Vis a quasi-isomorphism. In view of [34,

II.1.2.4.4] we derive thatΩ(P a• )!!/V → ΩP•/V is a quasi-isomorphism,i.e. m⊗V ΩP•/V → ΩP•/V

is a quasi-isomorphism. Sincem is flat andΩP•/V → ΩP•/V ⊗P• B!! = LB!!/V is a quasi-isomorphism, we get the desired conclusion.

7.1.7.1.11. In view of proposition 7.1.8 we haveLa(V a×A)!!/V×A!!≃ 0 in D•(V a × A). By

this, transitivity and localisation ([34, II.2.3.1.1]) wederive thatLaB/A → LaB!!/A!!is a quasi-

isomorphism for allA-algebrasB. If A andB are exact (e.g. if m is flat), we conclude fromproposition 7.1.10 that the natural mapLB/A → LB!!/A!!


7.1.7.1.12. Finally we want to discuss left derived functors of (the almost version of) somenotable non-additive functors that play a role in deformation theory. LetR be a simplicialV -algebra. Then we have an obvious functorG : D•(R) → D•(Ra) obtained by applyingdimension-wise the localisation functor. LetΣ be the multiplicative set of morphisms ofD•(R)that induce almost isomorphisms on the cohomology modules.An argument as in section 2.4shows thatG induces an equivalence of categoriesΣ−1D•(R)→ D•(Ra).

7.1.7.1.13. Now letR be aV -algebra andFp one of the functors⊗p, Λp, Symp, Γp defined in[34, I.4.2.2.6].

Lemma 7.1.14.Let φ : M → N be an almost isomorphism ofR-modules. ThenFp(φ) :Fp(M)→ Fp(N) is an almost isomorphism.

Proof. Let ψ : m ⊗V N → M be the map corresponding to(φa)−1 under the bijection (2.2.4).By inspection, the compositionsφ ψ : m ⊗V N → N andψ (1m ⊗ φ) : m ⊗V M → Mare induced by scalar multiplication. Pick anys ∈ m and lift it to an elements ∈ m; defineψs : N →M by n 7→ ψ(s⊗ n) for all n ∈ N . Thenφ ψs = s · 1N andψs φ = s · 1M . Thiseasily implies thatsp annihilatesKerFp(φ) andCokerFp(φ). In light of proposition 2.1.7(ii),the claim follows.

7.1.7.1.15. LetB be an almostV -algebra. We define a functorF ap on B-Mod by letting

M 7→ (Fp(M!))a, whereM! is viewed as aB!!-module or aB∗-module (to show that these

choices define the same functor it suffices to observe thatB∗ ⊗B!!N ≃ N for all B∗-modules

N such thatN = m ·N). For allp > 0 we have diagrams :

R-ModFp //

R-Mod

Ra-Mod

Fap //

OO

Ra-Mod

OO(7.1.16)


where the downward arrows are localisation and the upward arrows are the functorsM 7→ M!.Lemma 7.1.14 implies that the downward arrows in the diagramcommute (up to a naturalisomorphism) with the horizontal ones. It will follow from the following proposition 7.1.18that the diagram commutes also going upward.

7.1.7.1.17. For anyV -moduleN we have an exact sequenceΓ2N → ⊗2N → Λ2N → 0. Asobserved in the proof of proposition 2.1.7, the symmetric groupS2 acts trivially on⊗2m andΓ2m ≃ ⊗2m, soΛ2m = 0. Also we have natural isomorphismsΓpm ≃ m for all p > 0.

Proposition 7.1.18.LetR be a commutative ring andL a flatR-module withΛ2L = 0. Thenfor p > 0 and for allR-modulesN we have natural isomorphisms

Γp(L)⊗R Fp(N)∼→ Fp(L⊗R N).

Proof. Fix an elementx ∈ Fp(N). For eachR-algebraR′ and each elementl ∈ R′ ⊗R L weget a mapφl : R′⊗RN → R′⊗RL⊗RN by y 7→ l⊗y, hence a mapFp(φl) : R

′⊗RFp(N) ≃Fp(R

′ ⊗R N) → Fp(R′ ⊗R L ⊗R N) ≃ R′ ⊗R Fp(L ⊗R N). For varyingl we obtain a

map of setsψR′,x : R′ ⊗R L → R′ ⊗R Fp(L ⊗R N) : l 7→ Fp(φl)(1 ⊗ x). According tothe terminology of [46], the system of mapsψR′,x for R′ ranging over allR-algebras forms ahomogeneous polynomial law of degreep from L to Fp(L ⊗R N), so it factors through theuniversal homogeneous degreep polynomial lawγp : L → Γp(L) . The resultingR-linearmapψx : Γp(L) → Fp(L ⊗R N) dependsR-linearly onx, hence we derive anR-linear mapψ : Γp(L)⊗RFp(N)→ Fp(L⊗RN). Next notice that by hypothesisS2 acts trivially on⊗2L soSp acts trivially on⊗pL and we get an isomorphismβ : Γp(L)

∼−→ ⊗pL. We deduce a natural

map(⊗pL) ⊗R Fp(N) → Fp(L ⊗R N). Now, in order to prove the proposition for the caseFp = ⊗p, it suffices to show that this last map is just the natural isomorphism that “reordersthe factors”. Indeed, letx1, ..., xn ∈ L andq := (q1, ..., qn) ∈ Nn such that|q| :=

∑i qi := p;

thenβ sends the generatorx[q1]1 · ... ·x[qn]n to

(p

q1,...,qn

)·x⊗q11 ⊗ ...⊗x⊗qnn . On the other hand, pick

anyy ∈ ⊗pN and letR[T ] := R[T1, ..., Tn] be the polynomialR-algebra inn variables; write(T1 ⊗ x1 + ... + Tn ⊗ xn)⊗p ⊗ y = ψR[T ],y(T1 ⊗ x1 + ... + Tn ⊗ xn) =

∑r∈Nn T r ⊗ wr with

wr ∈ ⊗p(L ⊗R N). Thenψ((x[q1]1 · ... · x[qn]n ) ⊗ y) = wq (see [46, pp.266-267]) and the claimfollows easily. Next notice thatΓp(L) is flat, so that tensoring withΓp(L) commutes with takingcoinvariants (resp. invariants) under the action of the symmetric group; this implies the assertionfor Fp := Symp (resp.Fp := TSp). To deal withFp := Λp recall that for anyV -moduleMandp > 0we have the antisymmetrizer operatoraM :=

∑σ∈Sp

sgn(σ)·σ : ⊗pM → ⊗pM and asurjectionΛp(M)→ Im(aM) which is an isomorphism forM free, hence forM flat. The resultfor Fp = ⊗p (and again the flatness ofΓp(L)) then givesΓp(L)⊗Im(aN) ≃ Im(aL⊗RN), hence

the assertion forFp = Λp andN flat. For generalN let F1∂→ F0

ε→ N → 0 be a presentation

with Fi free. Definej0, j1 : F0 ⊕ F1 → F0 by j0(x, y) := x + ∂(y) andj1(x, y) := x. Byfunctoriality we derive an exact sequence

Λp(F0 ⊕ F1)//// Λp(F0) // Λp(N) // 0

which reduces the assertion to the flat case. ForFp := Γp the same reduction argument works aswell (cp. [46, p.284]) and for flat modules the assertion forΓp follows from the correspondingassertion forTSp.

Lemma 7.1.19.LetA be a simplicial almost algebra,L,E andF threeA-modules,f : E → Fa quasi-isomorphism. IfL is flat orE, F are flat, thenL⊗A f : L⊗A E → L⊗A F is a quasi-isomorphism.

Proof. It is deduced directly from [34, I.3.3.2.1] by applyingM 7→M!.


7.1.7.1.20. As usual, this allows one to show that⊗ : Hot•(A)×Hot•(A)→ Hot•(A) admits

a left derived functorL

⊗ : D•(A) × D•(A) → D•(A). If R is a simplicialV -algebra then wehave essentially commutative diagrams

D•(R)× D•(R)L

⊗ //

D•(R)

D•(Ra)× D•(Ra)

L

⊗ //

OO

D•(Ra)

OO

where again the downward (resp. upward) functors are induced by localisation (resp. byM 7→M!).

7.1.7.1.21. We mention the derived functors of the non-additive functorFp defined above inthe simplest case of modules over a constant simplicial ring. Let A be a (commutative)V a-algebra.

Lemma 7.1.22. If u : X → Y is a quasi-isomorphism of flats.A-modules thenF ap (u) :

F ap (X)→ F a

p (Y ) is a quasi-isomorphism.

Proof. This is deduced from [34, I.4.2.2.1] applied toN(X!) → N(Y!) which is a quasi-isomorphism of chain complexes of flatA!!-modules. We note thatloc. cit. deals with a moregeneral mixed simplicial construction ofFp which applies to bounded above complexes, butone can check that it reduces to the simplicial definition forcomplexes inC•(A!!).

7.1.7.1.23. Using the lemma one can constructLF ap : D•(s.A) → D•(s.A). If R is a V -

algebra we have the derived category version of the essentially commutative squares (7.1.16),relatingLFp : D•(s.R)→ D•(s.R) andLF a

p : D•(s.Ra)→ D•(s.Ra).


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INDEX

(V,m), m : Basic setup, 2.1.2.1.1B(m), 3.4.3.4.7IB/A, 2.5.21K+, 5.1.5.1.11K+h,K+sh,Ksh,Ka, 5.1.12, 5.2.16Ma, 2.2.2.2.2R-Mod,R-Alg (for a ringR), 2.2.2.2.1V -Mon : category ofV -monoids, 3.4.3.4.2V a-Alg,A-Mod,A-Alg (for an almost algebraA)

, 2.2.2.2.7V a-Alg.Morph, V a-Alg.Mod, 2.5.21V a-Mod, (V,m)a-Mod, 2.2.2.2.2X ∗ φ, ψ ∗X , 2.5.2.5.5alHomA(M,N) : Almost homomorphisms, 2.2.2.2.10B!!, 2.2.2.2.23Desc(C , Y/X), 3.3.3.3.18ExmonC (B, I), ExalC (B, I) , 2.5.2.5.5, 2.5.2.5.7Gm,R(n), 4.2.4.2.1Γ+, 5.1.20Idemp(R), 3.3.3.3.36I!∗, 2.5.19LanB/A : Analytic cotangent complex, 6.1.23

LX/Y : Cotangent complex for formal schemes, 6.2.3L+d(X)/d(Y) : Cotangent complex for Zariski-Riemann

spaces, 6.2.23M!, 2.2.2.2.19Ω+

d(X)/K , 6.3Set, 2.2.2.2.1TorAi (M,N), alExtiA(M,N), 2.4.2.4.8ExalA,A†-Uni.Mod, 2.5.2.5.7, 2.5.2.5.8Exmon, ExunC , ExalC , 2.5.2, 2.5.2.5.4µ(F ), 5.2.5.2.3B, 3.4B-monoid(s), 3.4.3.4.2ΦA : Frobenius endomorphism of, 3.4.3.4.6B-Mon : category of, 3.4.3.4.2Frobenius nilpotent, 3.4.8invertible morphism up toΦm of , 3.4.8

B/Fp, B-Alg/Fp, B-Mon/Fp, Ba-Alg/Fp , 3.4.3.4.5B-Alg : category ofB-algebras, 3.4.3.4.2B-Mod : category ofB-modules, 3.4.3.4.1Ba-Mod, Ba-Alg, Ba-Mon, 3.4.3.4.3C , C o, s.C , 1X , D(C ) , 2.2,2.2.2.2.1C -groupoid(s), 4.5.4.5.1X0/G : quotient by an action of a, 4.5.5with trivial automorphisms, 4.5.4.5.1

Gn, G∞, 4.2.4.2.6, 4.3.9IA(M),EM (m0),EM (m0), 2.3.2Mc(A), 2.3.2.3.1Mi, Mi,fl, Mi,proj , 3.2.19χP (X), ψP (X), 4.3.4.3.7det(1P + φ), 4.3.4.3.1ηB,C , 4.1.13f.rkA(P, p), 4.4.4.4.13ExtpR(E,F ), 2.5.2.5.17HomOX

(F , ∗), 5.4.5.4.10

MndX , Z-ModX , 5.4.5.4.1Div(K+a) : Fractional ideals, 5.1.5.1.16µA, σM/A, 2.2.2.2.5ω(R, u′), 3.2.19ω(B, f0, u), ω(B, f0) , 3.2.3.2.7, 3.2.3.2.9ωP/A, 2.4.2.4.21ρE+/K+ , 5.5.5.5.1D−(A-Mod)∧, 6.1.6.1.13τe, τw, 3.3.3.3.36u.Et

o, 3.3.3.3.38

ζP , 4.1.1eB/A, 3.1.4uM , 4.4.29Bud(n, d,R) : n-buds of a formal group, 4.2.4.2.2N (A) : Spectrum of seminorms of aK-algebra, 6.4.10(n-)pseudo-coherent complex(es), 6.1.6.1.15

D(R-Mod)coh : derived category of, 6.1.6.1.16θM|N , 2.2.2.2.5

Adic space(s), 6.2.6.2.17etale morphism of, 6.2.6.2.19d+Y/X : discriminant of an, 6.4.6.4.9def(f) : defect of an, 6.4.6.4.24

affinoid, 6.2.6.2.17morphism locally of finite type of, 6.2.6.2.18smooth, unramified morphism of, 6.2.6.2.19

Affinoid ring(s), 6.2.6.2.12Spa(A) : adic spectrum of an, 6.2.6.2.16completion of, 6.2.6.2.13

Almost algebra(s), 2.2.2.2.5, 2.2.2.2.7DerA(B,M) : derivations of an, 2.5.21LB/A : cotangent complex of an, 2.5.20

transitivity of the , 2.5.30ΩB/A : relative differentials of an , 2.5.21nil(A) : nilradical of an, 4.1.18(faithfully) flat, 3.1.1(uniformly) almost finite, 3.1.1(weakly) unramified, 3.1.1(weakly) etale, 3.1.1w.Et(A), Et(A) : category of, 3.2.3.2.16

almost finite projective, 3.1.1almost projective, 3.1.1TrB/A : trace morphism of an , 4.1.4.1.7DB/A : different ideal of an , 4.1.22dB/A : discriminant of an, 6.4.6.4.2tB/A, τB/A : trace form of an , 4.1.4.1.12

exact, 2.2.25ideal in an, 2.2.2.2.5reduced, 4.1.18strictly finite, 3.1.1

Almost isomorphism, 2.1.2.1.3Almost module(s), 2.2.2.2.2M ⊗A N : tensor product of, 2.2.2.2.11M∗ : dual of an, 2.4.18M∗ : almost element(s) of an, 2.2.2.2.8, 2.2.2.2.9M -regular, 4.1.21

150


AnnA(M) : annihilator of an, 2.2.2.2.5Λ-nilpotent endomorphism of an , 4.3.4.3.1EM/A : evaluation ideal, 2.4.18Un(A) : uniformly almost finitely generatedFi(M) : Fitting ideals of a, 2.3.22

Un(A) : uniformly almost finitely generated , 2.3.9evM/A : evaluation morphism of an, 2.4.18(faithfully) flat, 2.4.4almost finitely generated , 2.3.9almost finitely generated projective, 2.4.4almost finitely presented , 2.3.9almost projective, 2.4.4Split(A,P ) : splitting algebra of an, 4.4.1dA(P, b) : discriminant of a bilinear form on a,

6.4.1f.rkA(P ) : formal rank of an , 4.3.9trP/A : trace morphism of an , 4.1.1of almost finite rank , 4.3.9of constant rank , 4.3.9of finite rank , 4.3.9

finitely generated, 2.3.9finitely presented, 2.3.9invertible, 2.4.18reflexive, 2.4.18strictly invertible , 4.4.29that admits infinite splittings , 4.4.15

Almost zero module, 2.1.2.1.3

bicovering morphism in a site, 3.4.3.4.16

Cauchy product, 2.3.2.3.5Classical limit, 2.1.2ConditionsA andB, 2.1.2.1.6

Decomposition subgroup, 5.2.5.2.3Deeply ramified system of formal schemes, 6.3.5

f-adic ring(s), 6.2.6.2.10Cont(A) : continuous valuations of an, 6.2.6.2.15topologically finite type map of, 6.2.6.2.11ideal of definition of an, 6.2.6.2.10ring of definition of an, 6.2.6.2.10

Inertia subgroup, 5.2.5.2.3

Log structure(s), 5.4.5.4.4logX : category of, 5.4.5.4.4regular, 5.4.5.4.9

Monoid(s), 5.1.5.1.26Mgp : group associated to an abelian, 5.1.5.1.29Spec(M) : spectrum of a, 5.1.5.1.27Mnd : category of commutative, 5.1.5.1.29convex submonoid of a, 5.1.5.1.26free, 5.1.28

basis of a, 5.1.28ideal, prime ideal in a, 5.1.5.1.27integral, 5.1.28

Ordered abelian group(s), 5.1.20

Spec(Γ) : spectrum of an, 5.1.20c.rk(Γ), rk(Γ) : convex and rational rank of an,

5.1.20convex subgroup of an, 5.1.20

Pre-log structure(s), 5.4.5.4.2ΩX/Z(logM) : logarithmic differentials of a, 5.4.5.4.11pre-logX : category of, 5.4.5.4.2

Prufer Domain, 5.1.12

Simplicial almost algebra(s), 7.1.7.1.1exact, 7.1.7.1.3modules over a, 7.1.7.1.1

derived category of , 7.1.7.1.2quasi-isomorphism of , 7.1.7.1.3

Standard setup, 5.1.5.1.15

Tensor category, 2.2.2.2.5(left, right, unitary) module in a , 2.2.2.2.5algebra in a, 2.2.2.2.5monoid in a, 2.2.2.2.5ExmonC (B, I) : extensions of a , 2.5.2

Valuation ring(s), 5.1.5.1.11total log structure on a, 5.4.5.4.14

Valuation(s), 5.1.5.1.1Gauss, 5.1.4rank of a, 5.1.5.1.23

Value group, 5.1.5.1.1Valued field(s), 5.1.5.1.1Kt : maximal tame extension of a, 5.2.16deeply ramified, 5.6.1extension of a, 5.1.5.1.3d+L/K : discriminant of an algebraic, 6.4.6.4.5

valuation topology of a, 5.1.21

arXiv:math/0201175v1 [math.AG] 18 Jan 2002 · such case E+ is a G-torsor for the e´tale topology...

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