Introductionmath.colorado.edu/~wigi1604/diffspace.pdfspaces locally isomorphic to zero loci of...

$: Introductionmath.colorado.edu/~wigi1604/diffspace.pdfspaces locally isomorphic to zero loci of \Taylor closed" ideals in open subsets of Rn. Our approach is somewhat more abstract$
ON DIFFERENTIABLE AND ANALYTIC SPACES

W. D. GILLAM

Abstract. A general theory of “differentiable spaces” is proposed. The theory includes(at least) the theories of real and complex analytic spaces, as well as Spallek’s differen-tiable spaces as special cases.

1. Introduction

This note is based on my interpretation of the book C∞-Differentiable Spaces [DS] inwhich the authors develop a theory of “differentiable spaces” along lines suggested byK. Spallek in a series of papers from 1969-1974 (see the references to [DS]). The essentialpoints of this approach are outlined in the paper [Nav] of J. A. Navarro Gonzalez. Amongother things, the authors describe the category DS of “differentiable spaces.” The categoryDS is a full subcategory of the category LRS/R of locally ringed spaces over R enjoyingthe following properties:

DS1. For each n ∈ N, the space Rn, with the metric topology and sheaf C∞ of smoothreal-valued functions, is in DS and Rn represents the presheaf

DSop → Sets

(X,OX) 7→ OX(X)n.

DS2. For (X,OX) ∈ LRS/R and any open cover Ui of X, we have (X,OX) ∈ DS iff(Ui,OX |Ui) ∈ DS for all i.

DS3. DS has all finite inverse limits.DS4. DS contains SpecR[x]/xn for any n ∈ N.

The conditions (DS1) and (DS2) ensure that DS contains all smooth manifolds andthat the product of smooth manifolds in DS is their “usual” product. The category DSin fact contains a rich theory of non-reduced spaces of which (DS4) is but a shadow.

Notice that the category RAS of real analytic spaces enjoys similar properties, as doesthe category AS of (complex) analytic spaces (replacing R with C and “smooth” with “an-alytic” where appropriate). Here we put forth a general setup for studying “certain kindsof functions” on open subsets of kn, k a topological field, which includes the constructionsof DS, RAS, AS, etc. as special cases.

To appreciate the subtlety of the theory, we should begin by pointing out an interestingdifference between DS and AS. The forgetful functor AS→ LRS/C preserves equalizers,while the forgetful functor DS→ LRS/R does not. Note that neither forgetful functor canpreserve products because of (DS1).1 I conjecture that there is no full subcategory category

Date: September 18, 2013.1This is because (DS1) implies that the smooth manifold R2 is the product of the real line R with itself

in DS, but the product of R with itself in LRS/R is something altogether different. They are the same1

2 W. D. GILLAM

DS ⊆ LRS/R satisfying (DS1) with the property that DS has equalizers commuting withthe inclusion DS → LRS/R. See §3.11. In any case, even if there were such a category,there are various senses in which this category would be “difficult” to work with, as thefollowing example suggests.

1.1. Example. Let f : R → R be a smooth function vanishing only at zero with zeroTaylor series at zero. The equalizer of 0, f : R ⇒ R in LRS/R is the locally ringed spacewith one point whose local ring A is the coequalizer of 0∗, f∗ : C∞0 → C∞0 , where C∞0 is theR-algebra of germs of smooth real valued functions at the origin in R. That is, A is thequotient of C∞0 by the ideal

I = g(f(x))− g(0) : g ∈ C∞0 (1.1.1)

=

n∑

i=1

hi · (gi f) : hi, gi ∈ C∞0

.

This ideal I depends on the choice of such an f . The local R-algebra A is very subtle.It is probably not noetherian and its isomorphism type as a local R-algebra presumablydepends on the choice of such an f . There are other delicate questions one can ask aboutA that are not easy to answer (§3.11). However, for most practical purposes (e.g. any“infinitesimal” study), this equalizer is way too “fine”. The approach taken here and in[DS] is to replace I by the larger ideal IT = (0)T of all flat functions: those g ∈ C∞0 withzero Taylor series at the origin. The quotient B := C∞0 /IT is nothing but the formal powerseries ring R[[x]] because a classical result of Borel ([Mal], I.4) says that the Taylor seriesat zero defines a surjective map of rings C∞0 → R[[x]], whose kernel is IT by definition(this holds in higher dimensions as well). The equalizer of 0, f in DS is the one pointspace with local ring R[[x]].

The above example gives the gist of the general theory. We allow only locally ringedspaces locally isomorphic to zero loci of “Taylor closed” ideals in open subsets of Rn. Ourapproach is somewhat more abstract and general than the one in [DS]. It is my opin-ion that this abstract approach is simpler for the usual reasons: One can focus on thepoints that are truly essential to the development of the theory without being distractedby various properties that may or may not be present in any particular manifestation.Here this means, among other things, that we will develop the theory of analytic spaces,differentiable spaces, etc. on an equal footing for as long as possible before concentratingon the special properties of any one of these particular theories. One interesting featureof our approach is that complex analytic spaces, in the usual sense (c.f. [GAGA, §1]), arisenaturally as the full subcategory of “locally finite type” spaces (§3.5) inside a larger cate-gory reminiscent of the categorical generalized analytic spaces and geometrical generalizedanalytic spaces of Kato and Usui [KU, §3.2].

I have endeavored to avoid any “analytic” theory in this presentation. For example, inour approach one does not need to know anything about the Frechet topology on the ringof smooth functions, nor does one need to know anything about Frechet modules over suchrings. Instead, one needs only a few basic facts about “continuous” modules over a localring with its m-adic topology—the basic theory can in fact be developed with nothing

on the level of topological spaces, but the local rings are radically different. Germs of smooth functions atthe origin in R2 is not the tensor product (over R) of two copies of the ring of germs of smooth functionsat the origin in R. Similar remarks hold for the analytic spaces C2 and C.

ON DIFFERENTIABLE AND ANALYTIC SPACES 3

more than the notion of the m-adic completion of a module over a local ring, though somemore “topological modules” machinery is necessary for a satisfactory theory of modules,coherence conditions, and differentials. We do not use the notion of topological tensorproduct of rings of smooth functions and Frechet modules over such rings as in [DS, 6]—this substantially simplifies, in my opinion, the construction of products and other inverselimits. The equivalence between the algebraic approach pursued here and the analyticapproach is provided by the “Spectral Theorem” of H. Whitney [Whi], [DS, 2.28], whichsays that our (purely algebraic) notion of “Taylor closed” (§2.5), in the case of ideals inthe sheaf of smooth real-valued functions on a manifold, is equivalent to being closed inthe Frechet topology. We do not need this fact anywhere in the present paper, which isentirely self-contained. In our algebraic approach the construction of the category DS andthe proofs of properties (DS1)-(DS4) above are a matter of “general nonsense.”

Acknowledgements. This research was partially supported by an NSF Postdoctoral Fel-lowship. I thank Chris Kottke for helpful comments and for directing me to some relevantliterature. I also thank Sam Molcho and Dinakar Muthiah for enlightening discussion ofthis work and for suggestions leading to numerous improvements to the exposition.

2. Generalities

To formulate the general results of this paper, we need several definitions. We considera fixed topological field k throughout. For concreteness, one may think of the cases k = Ror k = C with the metric topology. Beginning in §2.4 and at several other points in thepaper we will require various general results on m-adic completions of local rings and theirmodules. The necessary technical results are assembled in the Appendix. The reader witha basic knowledge of m-adic completion should be able to read the body of the paperindependently from the Appendix, but the reader who is not comfortable with completionmight perhaps begin by reading the first two or three sections of the Appendix.

2.1. Notation. A map of ringed spaces f = (f, f ]) : (X,OX)→ (Y,OY ) is a pair consist-ing of a continuous map f : X → Y and a map f ] : f−1OY → OX of sheaves of rings onX. The stalk of f ] at x ∈ X will be denoted fx : OY,f(x) → OX,x. A ringed space (X,OX)is locally ringed iff the stalk OX,x is a local ring for every x ∈ X. For a locally ringedspace X, the unique maximal ideal of OX,x will be denoted mx, or just m if x is clear fromcontext. A morphism of locally ringed spaces is a morphism of ringed spaces such thateach map fx is a local map of local rings (i.e. f(mf(x)) ⊆ mx). Given an open set U ⊆ Y

and an s ∈ OY (U), we let f∗s := f ](f−1s) ∈ OX(f−1U). Given an ideal I ⊆ OY , wewrite f−1I ⊆ OX as slight abuse of notation for the inverse image ideal : the ideal of OX

generated by f ](f−1I), where, here, f−1I ⊆ f−1OY is the usual sheaf-theoretic inverseimage.

2.2. Recollection. Let X be a locally ringed space over k. Recall [EGA Err I.1.8.1] thatthe natural map

HomLRS/k(X,Spec k[x1, . . . , xn]) → OX(X)n(2.2.1)

f 7→ (f∗x1, . . . , f∗xn)

4 W. D. GILLAM

is an isomorphism, so Spec k[x1, . . . , xn] represents the functor

(LRS/k)op → Sets(2.2.2)

X 7→ OX(X)n.

2.3. Zero loci. Suppose (X,OX) is a locally ringed space and I ⊆ OX is an ideal. Let

Z(I) := x ∈ X : Ix ⊆ mx.The subset Z(I) ⊆ X is closed because if Ix is not contained in mx, then Ix contains aunit ux, and that unit lifts to a unit u ∈ I(U) ∩ O∗X(U) on a neighborhood U of x, so Uis disjoint from Z(I). Note that the subspace Z(I) ⊆ X depends only on the images ofsections of I in the residue fields k(x). Let i : Z(I) → X be the inclusion and give Z(I)the sheaf of rings OZ(I) := i−1OX/I. The locally ringed space Z(I) is called the zero-locusof I. There is a natural monomorphism i : Z(I) → X of locally ringed spaces, whichis terminal among locally ringed spaces f : Y → X over X satisfying f−1I = 0 (inverseimage ideal sheaf notation of (2.1)) and there is an exact sequence

0→ I → OX → i∗OZ(I) → 0(2.3.1)

of OX modules.

Given a zero locus i : Z(I) → X and a zero locus j : Z(J) → Z(I), the compositionij : Z(J) → X is also a zero locus in X, namely Z(K), where K is the kernel of OX →(ij)∗OZ(J).

2.4. Taylor reduction. Let X be a locally ringed space. An OX module M is calledreduced iff, for all open U ⊆ X, the natural map M(U)→

∏x∈U Mx/mxMx taking a local

section to its images in all fibers is one-to-one. For x ∈ X, let

tx : Mx → Mx := lim←−

Mx/mnxMx

denote the mx-adic completion of the stalk Mx (c.f. the Appendix). For m ∈Mx, we call

tx(m) the Taylor series of m at x. Let tU : M(U)→∏

x∈U Mx be the product of the tx.We call tU (m) the Taylor series of m. We say that M is Taylor reduced iff, for all openU ⊆ X, the Taylor series map tU is one-to-one. We say that X is reduced (resp. Taylorreduced) iff OX is reduced (resp. Taylor reduced) as a module over itself.

Obviously reduced implies Taylor reduced. Of course, the sheaf of smooth (or analytic,or even continuous...) functions on a manifold is reduced. The structure sheaf of ananalytic space is Taylor reduced but not generally reduced. Any locally ringed space Xwhose stalks OX,x are noetherian (e.g. any locally noetherian scheme) is Taylor reduced

since the completion A → A of a noetherian local ring is faithfully flat; in particularinjective. In general, the categories we will construct will contain only Taylor reducedlocally ringed spaces.

When X is Taylor reduced (or even reduced), it is not necessarily true that tx : OX,x →OX,x is injective. For example, tx is not injective when X = (Rn, C∞) (unless n = 0), eventhough this X is reduced. The one point space x with “sheaf” of rings C∞x is the basicexample of a locally ringed space over R which is not Taylor reduced.

For any OX module M , we define the submodule M [ ⊆M of flat sections of M by

M [(U) := m ∈M(U) : tU (m) = 0,


so M is Taylor reduced iff M [ = 0. At each point x ∈ X, we have a containment of stalks

M [x ⊆ Ker(tx : Mx → Mx),(2.4.1)

but this is not generally an equality: When x is the origin, the f of (1.1) is a nonzero

element of the right hand side, but the left hand side is zero. The quotient M † := M/M [

is called the Taylor reduction of M . By (2.4.1) and Lemma 31, the map

Mx → M †x(2.4.2)

is an isomorphism for every x ∈ X. It follows immediately that M † is Taylor reducedand the projection M →M † is initial among OX module morphisms from M to a Taylorreduced OX module. Any Taylor reduced module M is annihilated by O[

X ⊆ OX , hence

may be regarded as a module over the Taylor reduction O†X of the structure sheaf.

Lemma 1. Let f : A → B be a local map of local sheaves of rings on a space X. For aB module M , let Mf denote M , regarded as an A module via restriction of scalars along

f . Then M [f ⊆ M [ with equality if f is surjective. In particular, Mf is a Taylor reduced

A module whenever M is a Taylor reduced B module and the converse holds when f issurjective.

Proof. Since f is local, we have fx(mA,x) ⊆ mB,x for every x ∈ X and hence we havemn

A,x(Mf )x ⊆ mnB,xMx for every n ∈ N. By Lemma 31, we have

Ker(tx : (Mf )x → (Mf )x) =⋂n

mnA,x(Mf )x

⊆⋂n

mnB,xMx

= Ker(tx : Mx → Mx).

If f is surjective, then all of the aforementioned containments are equalities. The result isnow clear from the definition of the submodule of flat sections.

2.5. Taylor closure. Let X be a locally ringed space, M ⊆ N a monomorphism of OX

modules. For each x ∈ X, give Mx the topology inherited from the mx-adic topology onNx (this is not generally the same as the mx-adic topology on Mx), and let

Mx = lim←−

Mx/(Mx ∩mnxNx)

be the completion of Mx in this topology. We have a commutative diagram with exactrows

0 // Mx

// Nx

tx

// (N/M)x //

tx

0

0 // Mx// Nx

// (N/M)x// 0

(2.5.1)

(see Remark 4.8 or [Mat, 8.1]). The Taylor closure MT ⊆ N is the submodule of Nwhose sections over U are those n ∈ N(U) such that tx(nx) ∈ Mx for all x ∈ U . ClearlyM ⊆MT . We say that M is Taylor closed in N iff M = MT and we say that M is Taylordense in N iff MT = N .

6 W. D. GILLAM

We will be most interested in these notions when M ⊆ N is the inclusion I ⊆ OX of anideal. In this case it is clear that the zero loci Z(I) and Z(IT ) agree on the level of spacessince the subspace Z(I) depends only on images of sections of I in residue fields (§2.3).

Lemma 2. For any submodule M ⊆ N , we have N/MT = (N/M)†. In particular, M isTaylor closed in N iff N/M is Taylor reduced.

Proof. We have surjections N → N/MT and N → (N/M)†, so it suffices to show that theyhave the same kernel. A local section n ∈ N(U) maps to zero in (N/M)†(U) iff its image nin (N/M)(U) has tx(nx) = 0 for all x ∈ U . By commutativity of (2.5.1) and exactness ofits bottom row, this is equivalent to saying tx(nx) ∈Mx for all x ∈ U , which is equivalentto n ∈MT (U) which is equivalent to saying n is in the kernel of N(U)→ (N/MT )(U).

Lemma 3. Let (X,OX) be a locally ringed space, I ⊆ OX an ideal. The zero locus Z(I)is Taylor reduced iff I is Taylor closed in OX .

Proof. Let i : Z(I) → X denote the inclusion. Recall the exact sequence

0→ I → OX → i∗OZ(I) → 0

of (2.3.1). Since i is a closed embedding of spaces, the stalks of i∗OZ(I) are given byOZ(I),x when x ∈ Z(I) ⊆ X and zero otherwise. From this and Lemma 1 it is clear thatZ(I) is Taylor reduced iff i∗OZ(I) is a Taylor reduced OX module, so the result followsfrom Lemma 2.

Lemma 4. For submodules I, J ⊆M , we have (IT + JT )T = (I + J)T .

Proof. It is enough to show that the corresponding quotientsM/(IT+JT )T andM/(I+J)T

are equal. Both are Taylor reduced by Lemma 2, so its suffices to show that both have thesame maps to any Taylor reduced module R. For any submodule K ⊆ M Lemma 2 andthe universal property of Taylor reduction together imply that a map M → R kills K iffit kills KT . Repeatedly applying this observation, we see that giving a map from eitherM/(IT + JT )T or M/(I + J)T to R is the same thing as giving a map M → R that killsboth I and J .

Lemma 5. Let X,Y be locally ringed spaces with X Taylor reduced, I ⊆ OY an ideal,f : X → Y a map of locally ringed spaces. Then f−1I = 0 iff f−1IT = 0.

Proof. We always have I ⊆ IT , so the only difficulty is to prove f−1IT = 0 when f−1I = 0.Consider some open U ⊆ X and some g ∈ (f−1IT )(U). We want to show that f ](g) is zeroin OX(U). Since X is Taylor reduced, it suffices to show that tx((f ]g)x) = tx(fx(gf(x)))

is zero in OX,x for every x ∈ U . We have a commutative diagram

If(x)

// OY,f(x)

tf(x)

fx // OX,x

tx

If(x) // OY,f(x)fx // OX,x

(2.5.2)

where the composition of the top horizontal arrows is zero since f−1I = 0, so the composi-tion of the bottom arrows is also zero by naturality of completion. Since g ∈ (f−1IT )(U),we know tf(x)(gf(x)) ∈ If(x) for all x ∈ U , so the desired vanishing follows from commuta-tivity of (2.5.2).


In light of Lemma 3, we can say that (X,OX) is Taylor reduced iff the zero ideal isTaylor closed. For any locally ringed space X, if we let I := (0)T be the Taylor closure ofthe zero ideal, then Z(I) is Taylor-reduced, and the inclusion Z(I)→ X is an isomorphismon the level of spaces and is terminal among maps from Taylor-reduced spaces to X.

The behaviour of Taylor closure and Taylor reduction under tensor products and inverseimages is very subtle. In the interest of simplicity we will put off discussing this for aslong as possible.

2.6. Remark. In the case of the sheaf of smooth functions on a manifold (or the structuresheaf of a differentiable space), Lemma 34 in the appendix shows that our notion of Taylorclosed ideal is the same as the one in [DS] (c.f. [DS Page 29, 2.16]).

2.7. Example. Let X = (Rn, C∞) and suppose f1, . . . , fm ∈ C∞(X) are real-analyticfunctions. It is a nontrivial theorem of Malgrange ([Mal], 6.1.1) that I := (f1, . . . , fm) isTaylor closed in C∞.

2.8. Questions. Taylor closure and reduction are the most mysterious operations in thispaper. For example, I do not know the answers to the following questions, nor can I findthem in [DS]. These questions are probably most meaningful if X is an O-space in oursense, or a differentiable space, but they make sense for any locally ringed space.

2.8.1. If I, J ⊆ OX are Taylor closed, does this imply that I + J is Taylor closed?

2.8.2. If I is Taylor closed, does this imply that I2 is Taylor closed? This question isimplicit at several points in [DS], but they don’t mention it explicitly. The answer istrivially “yes” if the underlying space Z(I) is a point, which is the only case discussed in[DS].

2.8.3. When is the inverse image of a Taylor closed ideal Taylor closed? This is the onlyquestion where I know of a nontrivial partial result: It is known that if π : X × Y → X isthe projection from a product of smooth manifolds and I ⊆ OX is Taylor closed, then sois π−1I ⊆ OX×Y .

2.8.4. If X ∈ LRS/k is Taylor reduced, is the sheaf of Kahler differentials ΩX/k Taylorreduced? We will return to this point in §4.

2.9. Continuous functions. For a topological space X, let

C(X, k) : U 7→ HomTop(U, k)

denote the sheaf of continuous functions to k on X, so (X, C(X, k)) is a reduced locallyringed space over k. Let Cn,0 := C(kn, k)0 denote the stalk of C(kn, k) at the origin (thering of germs of continuous functions to k near the origin), where kn has the producttopology. Since k is a topological field, kn is a homogeneous space under the translationaction, so we have a natural identification C(kn, k)x = Cn,0 for any x ∈ kn. The definitionof topological field ensures that rational functions are continuous and the topology on kn

refines the Zariski topology, so we obtain a natural morphism

e : (kn, C(kn, k)) → Spec k[x1, . . . , xn] =: k[x](2.9.1)

in LRS/k taking kn bijectively onto the set of k points of Spec k[x].

8 W. D. GILLAM

2.10. Algebras of functions. An algebra of functions O is a set On,0 → Cn,0 : n ∈ Nof injective local maps of local k algebras. In particular, O0,0 = k, On,0 contains theconstant functions and a function is invertible in On,0 iff it does not vanish at the origin.Such an O determines a (reduced) sheaf of rings On ⊆ C(kn, k) on each kn by declaring(f : U → k) ∈ C(kn, k)(U) to be in On(U) iff the stalk (germ) fx ∈ C(kn, k)x = Cn,0 lies inthe subring On,0 for every x ∈ U . Evidently we have a natural identification On,x = On,0

for each x ∈ kn and the space (kn,On) is again a homogeneous space under translation.We also have an epimorphism2

i : (kn, C(kn, k)) → (kn,On)(2.10.1)

of locally ringed spaces over k.

An algebra of functions O is called admissible iff it satisfies the following conditions:

A1. On,0 contains the (germs at 0 of the) coordinate projections xi : kn → k. Thesexi are in the maximal ideal of On,0 since On,0 → Cn,0 is local, and for the samereason, On,0 contains the image of k[x](x) in Cn,0 (germs of rational functions) forevery n ∈ N. This implies that the map e of (2.9.1) factors (necessarily uniquely)as e = iw for some LRS/k morphism

w : (kn,On) → Spec k[x].

A2. The rings On are closed under composition (“pullback”) in the following sense:Functions f1, . . . , fm ∈ On(U) ⊆ C(kn, k)(U) determine a continuous map f =(f1, . . . , fm) : U → km. For any open V ⊆ km and any g ∈ Om(V ) ⊆ C(km, k)(V ),we require the continuous function gf : g−1(V ) ∩ U → k to be in On(g−1V ∩ U).

A3. The image of k[[x]]→ On,0 is dense in the m-adic topology on On,0.

Condition (A3) requires some discussion. The map w : k[[x]] → On,0 is the map onm-adic completions induced by the local map of local rings w : k[x](x) → On,0 of (A1). To

say that the image of w is dense means that for any a ∈ On,0 and any n ∈ N, there is ab ∈ k[[x]] such that a − w(b) ∈ mn. In practice, the map w will always be surjective, so(A3) will be obvious. We will typically make use of (A3) through the following:

Lemma 6. Let y be a point of kn and let f, g : On,y → A be two local maps of local

k-algebras which agree on the images of the coordinate functions xi in On,y. Then f = g.

Proof. Say y = (y1, . . . , yn), yi ∈ k. We have an isomorphism T : On,0 → On,y given bytranslation: T (f(x)) = f(x − y). Since T (xi) = xi − yi and f, g are k-algebra maps, wealso have f(xi − yi) = g(xi − yi), so, by replacing f, g by fT , gT , we can assume y is theorigin. Since the k-algebra maps f, g agree on the xi, they in fact agree on the image of

k[x](x) in On,0. The completions f , g therefore continuous ring maps which agree on the

image of k[x](x) in On,0. The image of k[x](x) in its completion k[[x]] is dense and A, like

any completion, is separated, so f and g agree on the image of k[[x]] in On,0 by Lemma 30.

But this image is dense by hypothesis (A3), so f = g by the same lemma.

2This word is always used in the categorical sense and is not to be confused with “surjective” whichusually means some underlying map of sets is an epimorphism.


2.11. Examples. For any topological field k, the images of k[x](x) in Cn,0 define an ad-missible algebra of functions. Suppose k is a complete metric field. The k algebras On,0 ofconvergent power series form an admissible algebra of functions. In this case, the map in(A3) is an isomorphism. When k = R, the R-algebras On,0 given by germs of smooth real-valued functions form an admissible algebra of functions. The map in (A3) is surjectiveby a fundamental result of Borel [Mal, I.4].

3. O-Spaces

Fix an admissible algebra of functions O (§2.10) for the remainder of the paper. Wewill write kn for the locally ringed space (kn,On), and we will write O for On when n isclear from context. In this section we define the category O-spaces and establish its basicproperties.

3.1. Local subcategories. A full subcategory C of LRS is called local iff, for any(X,OX) ∈ LRS, and any open cover Ui of X, we have (X,OX) ∈ C iff (Ui,OX |Ui) ∈ Cfor all i. A local subcategory of LRS/k is defined similarly. Every full subcategory C iscontained in a smallest local subcategory, defined in an obvious manner.

3.2. O-manifolds. The smallest local subcategory of LRS/k containing kn : n ∈ N iscalled the category of O-manifolds, denoted OM. It follows from (A2) that the forgetfulfunctor OM→ Top is faithful and that the natural map

HomLRS/k(X, (kn,On)) → OX(X)n(3.2.1)

f 7→ (f∗x1, . . . , f∗xn)

is an isomorphism for any X ∈ OM.

3.3. Models and O-spaces. Let U be an open subset of kn and let I ⊆ On|U be a Taylorclosed ideal. The locally ringed space Z(I) over k is called a model. By definition, thecategory of O-spaces, denoted OS, is the full subcategory of LRS/k consisting of those(X,OX) admitting a cover Ui such that each (Ui,OX |Ui) is isomorphic to a model inLRS/k. Notice that an open subspace of a model is a model, so OS is a local subcategoryof LRS/k. Since each (U,On|U) is reduced, the zero ideal is Taylor closed (Lemma 3), soZ(0) = (U,On|U) itself is a model, so every O-manifold is an O-space.

3.4. Coherence conditions. Let X be a ringed space. An OX module M is called quasi-coherent iff, locally on X, there is a surjection from a direct sum of copies of OX to M .Such a surjection is the same thing as a subset S ⊆ Γ(X,M) such that sx : s ∈ Sgenerates Mx as an OX,x module for every x ∈ X. If the direct sum (equivalently the setS) can be taken finite, then M is of locally finite type.

Let X be a locally ringed space. An OX module M is called almost quasi-coherent iff,locally on X, there is a map from a direct sum of copies of OX to M with Taylor denseimage (§2.5). If the direct sum can be taken finite, then M is said to be of almost locallyfinite type.

For example, the ideal I ⊆ C∞ of flat functions (smooth functions with zero Taylorseries at the origin) mentioned in Example 1.1 is of almost locally finite type, but it is(probably) not of locally finite type.

10 W. D. GILLAM

There are certain reasonable hypotheses under which all OX modules are quasi-coherent,as we will now explain.

Definition 1. For a ringed space X, a point x ∈ X, and a neighborhood U of x, a bumpfunction on U is a section b ∈ OX(U) with stalk bx = 1 ∈ OX,x whose support

Supp b = u ∈ U : bu 6= 0

is a closed subset of X.3 We say that X has bump functions at x iff there is a cofinalsystem of neighborhoods U of x for which there is a bump function b ∈ OX(U). We saythat X has bump functions iff it has bump functions at x for every x ∈ X.

Lemma 7. Suppose X is a ringed space with bump functions. Then for any OX moduleM , any x ∈ X, and any m ∈ Mx, there is a global section m ∈ Γ(X,M) with stalkmx = m.

Proof. Lift m ∈ Mx to m′ ∈ M(U) on a neighborhood U of x. After shrinking U andrestricting m′ if necessary, we can assume there is a bump function b ∈ OX(U). ClearlySupp(bm′) ⊆ Supp b, so (bm′)x = 0 for x ∈ U \ Supp b. By the sheaf property for M , thesections bm′ ∈ OX(U) and 0 ∈ OX(U \ Supp b) glue to a global section m ∈M(X) whichis as desired since

mx = (bm′)x = bxm′x = m′x = m.

Lemma 8. Suppose X is a ringed space which locally has bump functions. Then any OX

module M is quasi-coherent.

Proof. The question is local so we can assume X has bump functions. For each x ∈ Xand m ∈ Mx we can find, by the previous lemma, a global section s(m,x) ∈ M(X) withs(m,x)x = m. The subset S = s(m,x) : x ∈ X,m ∈ Mx of Γ(X,M) clearly witnessesthe quasi-coherence of M . That is, ⊕SOX → M is surjective, since this can be checkedon stalks.

3.5. Finiteness conditions. A map f : X → Y of O-spaces is locally finite type (resp.almost locally finite type) iff, locally on X and Y , there is an open subspace U ⊆ kn suchthat f factors as the zero locus X = Z(I) → Y × U of a locally finite type (resp. almostlocally finite type) Taylor closed ideal I of Y ×U followed by the projection π1 : Y ×U → Y .By shrinking further if necessary, one may assume here that I is generated by finitely manyof its global sections f1, . . . , fm (resp. I is the Taylor closure of the ideal generated by thefi). The product Y ×U here is of course the product in the category of O-spaces. We willsee that this product exists in Lemma 20.

In particular, we say that an O-space X is of (almost) locally finite type iff the mapfrom X to k0 = Spec k is of (almost) locally finite type—i.e. X is locally isomorphic tothe zero locus Z(I) ⊆ U of a Taylor closed ideal generated by finitely many functionsf1, . . . , fm ∈ On(U) (resp. the Taylor closure of the ideal generated by such f1, . . . , fn) forsome open subset U ⊆ kn.

3If i : U → X denotes the inclusion and i! : Ab(U)→ Ab(X) is the pushforward with proper supportfunctor, then a bump function on U is a section b ∈ (i!i

−1M)(U) ⊆M(U) with bx = 1.


3.6. Differentiable spaces. Suppose k = R with the metric topology, and O is theadmissible algebra of germs of smooth real valued functions on Rn at the origin (§2.11).The sheaves On,0 on Rn are then the usual sheaves C∞ of smooth real valued functions.These O-spaces are called differentiable spaces. In this setting, any point of Rn has acofinal system of neighborhoods isomorphic to Rn, so one may take as models the locallyringed spaces Z(I) ⊆ Rn, as I ranges over Taylor-closed ideals of C∞.

The space (Rn, C∞) has bump functions, hence so does Z(I) for any ideal I ⊆ C∞ (thisis obvious). In particular, all of the models Z(I) ⊆ Rn have bump functions, hence anydifferentiable space locally has bump functions, hence Lemma 8 implies

Lemma 9. For any differentiable space X, every OX module M is quasi-coherent.

3.7. First properties. Suppose P is a property of objects of LRS/k which is local, isinherited by zero loci of Taylor closed ideals, and is enjoyed by the spaces (kn,On). Thenit is clear that every X ∈ OS has property P. For example, here are some properties ofO-spaces easily obtained via this line of reasoning:

Proposition 10. Let X be an O-space. Then:

(1) The topological space X is locally isomorphic to a closed subset of an open subsetof kn.

(2) X is Taylor reduced.(3) Every point x ∈ X is a k point (the natural map from k to the residue field

k(x) := OX,x/mx is an isomorphism).(4) For every open U ⊆ X and every f ∈ OX(U), the function U → k given by

x 7→ f(x) is continuous. Here f(x) is the image of fx ∈ OX,x in the residue fieldk(x) = k.

Lemma 11. Every O-space is T1 (all points of the underlying space are closed).

Proof. Recall our tacit assumption that 0 is closed in k, which implies any point of k isclosed since k is a topological field, so translation is a homeomorphism. The property ofbeing T1 is inherited by products and closed subspaces, hence is enjoyed by any modelO-space. It is also local, hence is enjoyed by all O-spaces.

Lemma 12. Let Xi be a finite inverse limit system of O-spaces. The inverse limit of theXi taken in LRS/k coincides with the inverse limit of the Xi taken in RS/k.

Proof. This follows formally from (3) and (4) in Proposition 10. See [G, §3.2, Rem. 10].

The common inverse limit of the Xi mentioned in Lemma 12 is generally not an O-spaceand is usually of little interest to us.

Lemma 13. Suppose X ∈ LRS/k is Taylor-reduced and f1, f2 : (X,OX)→ (kn,On) aremaps of locally ringed spaces over k with f∗1xi = f∗2xi for all i. Then f1 = f2.

Proof. Certainly the compositions wf1, wf2 : X → Spec k[x1, . . . , xn] with the map w of(A1) agree by the discussion in (2.2), so f1 = f2 =: f on the level of spaces (w is monic

on spaces), so it is enough to show that f ]1, f]2 : f−1On → OX agree. Since X is Taylor-

reduced, it is enough to show that the maps f1,x, f2,x : On,f(x) → OX,x agree for anyx ∈ X, which is what we proved in Lemma 6.

12 W. D. GILLAM

Theorem 14. For any O-space X, the natural map

HomLRS/k(X, (kn,Ox)) → OX(X)n

f 7→ (f∗x1, . . . , f∗xn)

is bijective.

Proof. Every O-space is Taylor-reduced (10), so the map is injective by Lemma 13. Toshow it is surjective, consider f1, . . . , fn ∈ OX(X). Since we know there is at most onemap f : X → (kn,On) with f∗xi = fi, we can construct this map locally, so we canassume X = Z(I) ⊆ U ⊆ (km,Om) is a model. After shrinking to a smaller model ifnecessary, we can assume that there are lifts g1, . . . , gn ∈ Om(U) of the fi. The restrictionf of g := (g1, . . . , gn) : U → (kn,On) to X is then as desired.

Lemma 15. If X is an O-space and I ⊆ OX is a Taylor-closed ideal, then Z(I) is anO-space.

Proof. This is local, so we can assume X = Z(J) ⊆ U ⊆ kn is a model. Let i : Z(I) → Udenote the inclusion. Note Z(I) is Taylor reduced by Lemma 3, so the kernel K of thecomposition On|U → i∗Z(I) is Taylor closed and Z(I) is the zero locus, in (U,On|U), ofK (2.3), so it is a model.

Lemma 16. If X is an (almost) locally finite type O-space and I ⊆ OX is an (almost)locally finite type ideal, then Z(I) is an (almost) locally finite type O-space.

Proof. This is local, so we can assume X = Z(J) ⊆ U ⊆ kn with J equal to the idealgenerated by finitely many global functions f1, . . . , fm ∈ On(U) (resp. the Taylor closureof this ideal) and that I is generated by finitely many global functions g1, . . . , gk ∈ OX(X)(resp. the Taylor closure of this ideal). By shrinking further if necessary, we can assumethere are lifts h1, . . . , hk ∈ On(U) of the gi. Let K be the ideal of On|U generated by thefi and the hi (resp. the Taylor closure of this ideal). We then argue as in the previouslemma that Z(I) is isomorphic to Z(K).

3.8. Big and small induced structure. Let X be a locally ringed space, Z ⊆ X aclosed subspace of its underlying topological space. Both of the ideals

Ibig(Z) := f ∈ OX : tx(fx) = 0 for all x ∈ ZIsmall(Z) := f ∈ OX : f(x) = 0 for all x ∈ Z

= f ∈ OX : fx ∈ mx for all x ∈ Z

are clearly Taylor closed ideals of OX and we clearly have Ibig(Z) ⊆ Ismall(Z). If X is anO-space, then by Lemma 15, both Z(Ismall) and Z(Ibig) are O-spaces and we have closedembeddings

Z(Ismall(Z)) → Z(Ibig(Z)) → X.

For I = Ibig(Z) or I = Ismall(Z) it is clear that Iz ⊆ mz when z ∈ Z and that Ix = OX,x

when x ∈ X \Z, so we always have Z(Ismall(Z)) = Z(Ibig(Z)) = Z as closed subspaces ofX. However, the ideal I is often not even topologically quasi-coherent.

Example. Let Z = S1 be the unit circle in the complex analytic space C and U is aconnected neighborhood of a point z ∈ Z, then for either meaning of I we have I(U) = 0


since any analytic function vanishing on S1∩U vanishes on a convergent sequence of pointsof U , hence must be identically zero.

However, for differentiable spaces (§3.6), the big induced structure is rather well-behaved:

Lemma 17. Let X be a differentiable space, Z a closed subspace of the underlying topo-logical space of X . The ideal Ibig(Z) ⊆ OX is almost locally finite type.

Proof. The question is local, so we can assume X = Z(I) ⊆ Rn is a model. It is a standardfact that for any closed subspace Z ⊆ Rn, there is a smooth function f : U → R which isnon-vanishing away from Z whose Taylor series at any point of Z is identically zero. TheTaylor closure of the ideal generated by the image of f in OX(X) clearly coincides withIbig(Z).

I do not know whether Ismall(Z) is almost locally finite type for any closed subspaceZ ⊆ Rn, but I would not be surprised if this is true and well-known.

The O-space structure on Z given by Z(I (Z)) is often called the reduced-induced struc-ture. The O-space structure on Z given by Z(Ibig(Z)) is often called the big inducedstructure. The big induced structure is often very useful for differentiable spaces becauseof the aforementioned finiteness result and the following general representability lemma.

Lemma 18. The big induced structure Z(Ibig(Z)) on a closed subspace Z of an O-spaceX represents the presheaf taking U ∈ OSop to the set of f ∈ HomOS(U,X) which factorthrough Z on the level of topological spaces.

Proof. Set I := Ibig(Z). In light of the universal property of zero loci (§2.3), we arereduced to proving the following: f ] : f−1OX → OU kills f−1I iff f : U → X factorsthrough Z on the level of spaces. Suppose f doesn’t factor through Z on the level ofspaces. Then f(u) /∈ Z for some u ∈ U . But then If(u) = OX,f(u), so f ] certainly doesn’t

kill f−1I because the stalk fu of f ] at u takes 1 ∈ If(u) to 1 ∈ OU,u. This proves theeasy implication ( =⇒ ). Conversely, suppose f factors through Z on the level of spaces.To check that f ] kills f−1I we want to prove that for any open subset V ⊆ X and anys ∈ I(V ), the section g := f ](f−1s) ∈ OU (f−1(V )) is zero. Pick any point u ∈ f−1(V ).

The map f ]u : OX,f(u) → OU,u, being a local map of local rings, sits in a commutativediagram with the completion maps:

OX,f(u)fu //

tf(u)

OU,u

tu

OX,f(u)fu // OU,u

Since f(u) ∈ Z, sf(u) is in the kernel of tf(u) by definition of I. By commutativity we

conclude that gu = f ]u(sf(u)) is in the kernel the completion map tu for every u ∈ f−1(V ),hence g = 0 because O-spaces are Taylor reduced (Proposition 10(2)).

The above lemma illustrates one of the many differences between differentiable spaces,and, say, schemes. For the former, the big induced structure is well-behaved and the

14 W. D. GILLAM

reduced-induced structure is mysterious, whereas for the latter, the reduced-induced struc-ture is well-behaved (the ideal Ismall(Z) is always quasi-coherent and is coherent if X islocally noetherian), while the big induced structure is awful:

Example. When Z is the origin in the scheme X := A1k = Spec k[x], the big induced

structure on Z is the one point locally ringed space with structure “sheaf” given by the localring k[x](x). This locally ringed space is not a scheme, hence Ibig(Z) is not quasi-coherent.The locally ringed space Z in fact represents the presheaf LRS/k → Sets taking U tothe set of LRS/k morphisms f : U → X factoring set-theoretically through the origin.This doesn’t follow from Lemma 18 because not all schemes are Taylor reduced, but canbe seen directly as follows: We certainly have an LRS/k monomorphism g : Z → Xgiven by the inclusion of the origin on spaces and the isomorphism g−1OX → k[x](x) onstructure “sheaves”, so each LRS/k morphism f : U → Z gives an LRS/k morphismgf : U → X factoring set-theoretically through the origin. To go the other way, consideran LRS/k morphism f : U → X factoring set-theoretically through the origin. The mapf corresponds (§2.2) to a k-algebra map f : k[x]→ OU (U). To say that f factors throughthe origin means that for any u ∈ U , the preimge of mu under the composition of fand OU (U) → OU,u is the prime ideal (x). By the universal property of the localizationk[x] → k[x](x), we obtain a unique morphism of local k algebras fu : k[x](x) → OU,u

making the diagram of k algebras

k[x]f //

OU (U)

k[x](x)

fu // OU,u

commute. But this means that f(a)u is a unit for any a ∈ k[x] \ (x), so, since inversesare unique and OU is a sheaf, f(a) ∈ OU (U) is also a unit, hence f factors through thelocalization k[x] → k[x](x) via a map also abusively denoted f : k[x](x) → OU (U). Thismap f determines a map U → Z whose stalk at u ∈ U is the map fu above. We leaveit as an exercise for the reader to understand how Spec k[x](x) fits into this story and toprove that the analogous presheaf on schemes over k is not representable.

3.9. Inverse limits. Here we show that OS has inverse limits commuting with the un-derlying space functor OS→ Top.

Lemma 19. For any open subsets U ⊆ km, V ⊆ kn, the product

U × V = (U × V,Om+n|U × V )

is the product of U = (U,Om|U) and V = (V,On|V ) in OS.

Proof. Fix an O-space X. The universal property of open subspaces says that to give amap f : X → U is to give a map (f1, . . . , fn) : X → km factoring set theoretically throughU : (f1(x), . . . , fn(x)) ∈ U for all x ∈ X. Using a similar description of maps X → V andmaps X → U × V , together with Theorem 14, yields the result.

Lemma 20. The category OS has finite products commuting with OS→ Top.

Proof. By the usual gluing arguments [H II.3], [EGA I.3.2.6.2], [DS 7.6], it is enough toconstruct the product of models Z(I) ⊆ U ⊆ km and Z(J) ⊆ V ⊆ kn. By Lemma 19,


(U × V,Om+n|U × V ) is the product of U, V in OS. Let π1, π2 be the projections, letK ⊆ Om+n|U × V be the Taylor closure of the ideal K ′ := π−11 I + π−12 J , and let T :=Z(K) ⊆ U × V . This T is manifestly an O-space, and it is clear that T = Z(I) × Z(J)on the level of topological spaces (taking the Taylor closure of an ideal doesn’t effect thetopological space underlying its zero locus). For any O-space Y we compute

HomLRS/k(Y, T )

= f ∈ HomLRS/k(Y, U × V ) : f−1K = 0= f ∈ HomLRS/k(Y,U × V ) : f−1π−11 I + f−1π−12 J = 0= (f1, f2) ∈ HomLRS/k(Y,U)×HomLRS/k(Y, V ) : f−11 I = 0, f−12 J = 0= HomLRS/k(Y,Z(I))×HomLRS/k(Y,Z(J))

using the universal property of zero loci (2.3), the fact that Y is Taylor-reduced (Propo-sition 10(2)) and Lemma 5, so T is the product of Z(I) and Z(J) in OS.

Lemma 21. The category OS has equalizers commuting with OS→ Top.

Proof. Again, it is enough to treat the case of maps f, g : Z(I) ⇒ Z(J) for modelsZ(I) ⊆ U ⊆ (km,Om) and Z(J) ⊆ V ⊆ (kn,On). The inclusion Z(J) → (kn,On) is amonomorphism, so the equalizer of f, g is the same as the equalizer of their compositionswith this inclusion, so we reduce to the case of maps f, g : Z(I)→ (kn,On). Let K ⊆ OZ(I)

be the ideal generated by the global sections f∗xi−g∗xi ∈ Γ(Z(I),OZ(I)) and let KT be itsTaylor closure. For x ∈ Z(I) we have f(x) = g(x) ∈ kn iff (f∗xi−g∗xi)(x) = 0 ∈ k = k(x)for all i iff (f∗xi − g∗xi) ∈ mx for all i iff Kx ⊆ mx iff KT

x ⊆ mx iff x ∈ Z(KT ), so theclosed subspace Z(KT ) ⊆ Z(I) coincides with the equalizer of f, g on the level of spaces.Note Z(KT ) is an O-space by Lemma 15, and i : Z(KT ) → Z(I) is easily seen to be theequalizer of f, g using Lemma 5 and Theorem 14.

Theorem 22. The category OS has all finite inverse limits. The underlying space functorOS→ Top preserves finite inverse limits.

Proof. Combine Lemma 20 and Lemma 21.

3.10. O-ification. Here we describe the analog of the analytification of a scheme of locallyfinite type over C [GAGA, §2]. Let Sch/k denote the category of schemes of locally finitetype over k. For each X ∈ Sch/k, I claim there is an O-space XOS and a map XOS → Xof locally ringed spaces over k, called the O-ification of X, which is terminal among LRS/kmorphisms from an O-space to X. It follows immediately that the O-ification functor

Sch/k → OS

X 7→ XOS

preserves finite inverse limits.

In light of the universal property, it is enough to construct XOS locally, so we canassume X = Spec k[x1, . . . , xn]/(f1, . . . , fm). That is, X = Z(I) for I = (f1, . . . , fm) ⊆OSpec k[x1,...,xn]. By (A1), we can view f1, . . . , fm as elements of Γ(kn,On). Let J ⊆ On be

the ideal they generate, and let XOS := Z(JT ). The universal property of zero loci yieldsa map XOS → X which is easily seen to be as desired using Theorem 14 and Lemma 5.

Remark. In many settings of interest, we will have J = JT above. See Example 2.7.

16 W. D. GILLAM

There are many possible variants of the O-ification functor. For example, there is asmoothification XDS → X of a real analytic space X which is terminal among LRS/Rmorphisms from a differentiable space to X. It follows formally that smoothificationpreserves finite inverse limits.

3.11. Scholium. When describing the general idea of the theory of differentiable spaces(or O-spaces in general) to my colleagues, I am often asked why is the Taylor reduction stepnecessary? It is easier to answer the question if “necessary” is replaced by “desirable”. Myanswer would then be that the Taylor reduction greatly simplifies many of the structuresheaves and local rings one will encounter in the category of O-spaces: For example, Iwould rather not have to think about the rings A of Example 1.1—I prefer the ring R[[x]]for obvious reasons. Returning to the original question, I have to admit that I am notsure about whether the Taylor reduction step is truly necessary, but I believe that itmay be, for the following reason. The aforementioned ring A = C∞0 /I is scary to thinkabout as an abstract local R-algebra. Suppose, for example, that g1, g2 : C∞0 ⇒ A aretwo maps of local R-algebras such that g1(y) = g2(y) ∈ A (we use y for the coordinatefunction in the domain ring C∞0 to avoid confusion with the coordinate function x ∈ A).Recall that Lemma 30 implies that the compositions of g1, g2 with the completion map

t : A→ A = R[[x]] are equal because we have commutative diagrams

C∞0gi //

t

A

t

R[[y]]gi // R[[x]]

where the gi are continuous, R[y] ⊆ R[[y]] is dense, and R[[x]] is separated. But can weconclude that g1 = g2? I conjecture that we cannot. Let X be the one point space withlocal ring A. Assuming the conjecture, the natural map

HomLRS/R(X,R) → Γ(X,OX) = A

g 7→ g∗y

is not injective and the conclusions of Lemma 13 and Theorem 14 fail for this X. Recallthat X is the LRS/R equalizer of two maps R ⇒ R, so the conjecture precludes theexistence of a full subcategory i : D ⊆ LRS/R with the following properties:

(1) D contains the smooth manifold R(2) D has equalizers commuting with i(3) The conclusion of Lemma 13 (in the case k = R, n = 1) holds for X ∈ D.

Let us think about the conjecture for a moment. One “geometric” way to produce localR-algebra maps gi : C∞0 → A is to take smooth functions fi : R → R vanishing at 0 andlet gi be the composition of the pullback f∗i : C∞0 → C∞0 and the projection C∞0 → A. Ofcourse f∗i y = fi, so the condition g1(y) = g2(y) means f1 − f2 ∈ I ⊆ A. If we knew thatf1 = f2 in C∞0 (i.e. not just mod I), then the germs of f1 and f2 at the origin would beequal and it would be clear that g1 = g2, but we only know

f1 = f2 +

n∑i=1

hi · (gi f)


for some hi, gi ∈ C∞0 . It is not at all clear that this should imply that g1(h) = h f1 andg2(h) = h f2 are equal mod I for every h ∈ C∞0 . Besides, even if one knew this for such“geometric” gi, there are probably many “exotic” local R-algebra maps C∞0 → A that donot arise from the geometric construction.

4. Differential Theory

The purpose of this section is to develop an appropriate theory of differential formsfor reasonable O spaces. We will prove, under appropriate assumptions on the admissiblefunctions O, that differentials (in our sense) commute with inverse limits and with theO-ification functor of (3.10).

4.1. Scholium on Differentials. We begin by recalling some basic facts about Kahlerdifferentials.4 Fix a topological space X. Let An(X) be the category of sheaves of ringson X. Let FlAn(X) be the category of maps of sheaves of rings on X, whose morphismsare commutative diagrams. Let AnMod(X) be the category whose objects are pairs(A,M) consisting of a sheaf of rings A ∈ An(X) and an A module M . A morphism(f, g) : (A,M) → (B,N) in AnMod(X) is a pair consisting of a map f : A → B inAn(X) and an f linear map g : M → N—equivalently an A module homomorphismwhen N is regarded as an A module via restriction of scalars along f , or a B module

homomorphism M ⊗fA B → N . For an An(X) morphism f : A→ B, we let ΩB/A denote

the sheaf of Kahler differentials of B over A. That is, ΩB/A is the B module generated bysymbolds db, b ∈ B, subject to the relations d(f(a)) = 0 for all a ∈ A and

d(b1b2) = b1db2 + b2db1 for all b1, b2 ∈ B.(4.1.1)

The module ΩB/A is functorial in A→ B in the sense that it yields a functor

Ω : FlAn(X) → AnMod(X)

(f : A→ B) 7→ (B,ΩB/A).

For (A,M) ∈ AnMod(X), let A[M ] ∈ An(X) denote the trivial square zero extension ofA by M . As an A module, A[M ] = A⊕M . The multiplication in A[M ] is given by

(a,m)(a′,m′) := (aa′, am′ + a′m).(4.1.2)

The natural map a 7→ (a, 0) makes A[M ] an A algebra. Formation of A[M ] determines afunctor

[ ] : AnMod(X) → FlAn(X)

(A,M) 7→ (A→ A[M ])

4C.f. §II.1 in [Ill], [H, 2.8], [DS, §10]

18 W. D. GILLAM

which is right adjoint to Ω via the natural bijection below.

HomAnMod(X)((B,ΩB/A), (C,M)) = HomFlAn(X)((f : A→ B), (C → C[M ]))

(g : B → C, k : ΩB/A →M) 7→ Af //

gf

B

(g,b7→k(db))

C // C[M ]

(g, db 7→ hb) ←[ Af //

gf

B

(g,h)

C // C[M ]

In particular, taking B = C, g = Id above, we find that HomMod(B)(ΩB/A,M) isbijective with the set of An(X) morphisms (Id, h) : B → B[M ] making

Af //

f

B

(Id,h)

B(Id,0) // B[M ]

commute. If f is an epimorphism5 in An(X), then for any M the only possibility is h = 0,so we find ΩB/A = 0.

The following variant of the adjunction (Ω, [ ]) will be useful. Let LocAn(X) denotethe category whose objects are sheaves of rings onX with local stalks and whose morphismsare An(X) morphisms inducing local maps of local rings on stalks. Let FlLocAn(X)denote the category of morphisms in LocAn(X). Let LocAnMod(X) denote the cat-egory of pairs (A,M) where A ∈ LocAn(X) and M ∈ Mod(A) where a morphism(f, g) : (A,M) → (B,N) is a pair consisting of a LocAn(X) morphism f : A → B andan f linear map g : M → N . Then we also have an adjunction (Ω, [ ]) when we view Ωand [ ] as functors

Ω : FlLocAn(X) → LocAnMod(X)(4.1.3)

[ ] : LocAnMod(X) → FlLocAn(X).

For (A,M) ∈ LocAnMod, note that the unique maximal ideal m of Ax[Mx] is given by

m = mx ⊕Mx(4.1.4)

because for any u ∈ A∗x and any m ∈ M , we have (u,m)(u−1,−u−2m) = (1, 0) (c.f.(4.1.2)). It is clear from this that the functors Ω and [ ] make sense in this “local”setting and that the same formulas as in general setting yield the adjunction isomorphism

HomLocAnMod(X)((B,ΩB/A), (C,M))(4.1.5)

= HomFlLocAn(X)((f : A→ B), (C → C[M ])).

5We always use “epimorphism” in the categorical sense. A surjection in An(X) is an An(X) morphismf : A → B where f is an epimorphism of sheaves on X (surjective on stalks). A surjection in An(X) isalso an epimorphism in An(X), but not conversely: localizations and completions are epimorphisms inAn(X) but not generally surjections.


Since Ω is a left adjoint, it preserves direct limits in FlAn(X): If (fi : Ai → Bi)i is adirect limit system in FlAn(X) with limit f : A→ B (that is, A is the limit of the Ai, Bis the limit of the Bi, and f is the limit of the fi), then the natural map

lim−→

ΩBi/Ai⊗Bi B → ΩB/A(4.1.6)

is an isomorphism of B modules. In particular, if B = B1 ⊗A B2 is a tensor product, wehave:

ΩB/A = (ΩB1/A ⊗B1 B)⊕ (ΩB2/A ⊗B2 B)

ΩB/B2= ΩB1/A ⊗B1 B.

Similarly, Ω commutes with stalks in the sense that we have a natural isomorphism

ΩA/B,x = ΩAx/Bx(4.1.7)

of Ax modules for each point x ∈ X.

If f : B → C is a surjection of A algebras with kernel I, then in particular it is anepimorphism in An(X), so ΩC/B = 0 as we saw above. Furthermore, there is an exactsequence

I/I2 → ΩB/A ⊗B C → ΩC/A → 0(4.1.8)

of C modules, where the left arrow is give by [i] 7→ di⊗ 1, which is well-defined because

d(ij)⊗ 1 = (idj)⊗ 1 + (jdi)⊗ 1 = dj ⊗ f(i) + di⊗ f(j) = 0

when i, j ∈ I ⊆ B. A diagram

A→ B → C(4.1.9)

in An(X) yields an exact sequence

ΩB/A ⊗B C → ΩC/A → ΩC/B → 0(4.1.10)

of C modules.

The theory of the cotangent complex [Ill] associates, to an An(X) morphism A→ B, acomplex LB/A of flat B modules (supported in non-positive degrees) with H0(LB) = ΩB/A.The sequences (4.1.8), (4.1.10) are both special cases of long exact cohomology sequencesinduced by the short exact sequence

0→ LB/A ⊗B C → LC/A → LC/B → 0

of complexes of C modules associated to (4.1.9). If f : A → B is an epimorphism inAn(X), then we mentioned above that ΩB/A = 0. If, furthermore, f is flat, then in factLB/A is acyclic.

For a map of ringed spaces f : X → Y , we set ΩX/Y := ΩOX/f−1OY. Consider an

inverse limit system (fi : Xi → Yi)i of maps of ringed spaces with limit f : X → Y andprojections pi : X → Xi, qi : Y → Yi. Then the natural map

lim−→

p∗i ΩXi/Yi→ ΩX/Y(4.1.11)

is an isomorphism because the natural FlAn(X) morphism

lim−→

(p−1i f ]i : f−1q−1i OYi → p−1i OXi) → (f ] : f−1OY → OX)(4.1.12)

is an isomorphism by construction of inverse limits of ringed spaces. It is less obvious,but still true, that (4.1.11) is an isomorphism for an inverse limit system (fi : Xi → Y )i

20 W. D. GILLAM

of maps of locally ringed spaces (with the inverse limit taken in maps of locally ringedspaces6). In this case, (4.1.12) is not generally an isomorphism (for example because thesheaves of rings on the left might not have local stalks, in contrast to the sheaves of ringson the right). However, the maps

lim−→

p−1i OXi → OX

lim−→

q−1i OYi → OY

are localization morphisms [G], which are flat epimorphisms, so their sheaves of differentialsare zero (and their cotangent complexes are acyclic). One then concludes that (4.1.11) isan isomorphism by showing that the natural map from either side of (4.1.11) to the sheafof differentials of

lim−→

f−1q−1i OYi → OX

is an isomorphism using (4.1.10).7

4.2. Frechet differentials. Let f : X → Y be a map of locally ringed spaces. We refer to

the Taylor reduction (§2.4) Ω†X/Y of the Kahler differentials ΩX/Y as the sheaf of Frechet

differentials, or simply the differentials if there is no chance of confusion. If we restrict our

attention to Taylor reduced spaces, then we will see that the Frechet differentials Ω†X/Y

enjoy many of the same properties as the usual Kahler differentials. We will also see thatthe Frechet differentials behave nicely with respect to Taylor reduction, which makes thema useful substitute for the usual Kahler differentials in the setting of, say, differentiablespaces.

Proposition 23. Let A be a local sheaf of rings on a topological space X, M an A module.Then A[M ]† = A†[M †]. In particular, A[M ] is Taylor reduced iff both A and M are Taylorreduced.

Proof. For x ∈ X, we have A[M ]x = Ax[Mx]. Taylor reduction clearly commutes withfinite direct sums, so the point of the lemma is to see that the mx-adic topology on Ax[Mx],when viewed as the Ax module Ax ⊕Mx is the same as the m-adic topology on the localring Ax[Mx]. Recall from (4.1.4) that the unique maximal ideal of Ax[Mx] is given bym = mx ⊕Mx. It is easy to prove by induction on n that

mn = (mx ⊕Mx)n = mnx ⊕mn−1

x Mx,

which shows that the basic open neighborhoods of 0 ∈ Ax[Mx] in the m-adic topology arecofinal in the basic open neighborhoods mn

x(Ax ⊕Mx) = mnx ⊕ mn

xMx of 0 in the mx-adictopology on Ax ⊕Mx and vice-versa.

Let X be a topological space. Let LocAnMod(X) denote the full subcategory ofLocAnMod(X) (§4.1) consisting of those (A,M) where A and M are Taylor reduced.Let FlLocAn(X) denote the full subcategory of FlLocAn(X) consisting of those (f :A → B) where both A and B are Taylor reduced. By Proposition 23, we can view thetrivial square zero thickening functor [ ] of §4.1 as a functor

[ ] : LocAnMod(X) → FlLocAn(X).(4.2.1)

6It may be news to the reader that this limit even exists!7Technically, one also uses that the left map in (4.1.10) is an isomorphism when B → C is a localization

morphism, which is elementary, but also clear from the acyclicity of LC/B .


Similarly, since the Taylor reduction is always Taylor reduced (§2.4), we can view theFrechet differentials as a functor

FlLocAn(X) → LocAnMod(X).(4.2.2)

Proposition 24. The Frechet differentials functor (4.2.2) is left adjoint to (4.2.1).

Proof. Using the “usual” adjunction isomorphism (4.1.5), we obtain a sequence of naturalbijections

HomLocAnMod(X)((B,Ω†B/A), (C,M))

= HomLocAnMod(X)((B,ΩB/A), (C,M))

= HomFlLocAn(X)((f : A→ B), (C → C[M ]))

= HomFlLocAn(X)((f : A→ B), (C → C[M ])),

where the first equality holds because for any LocAn(X) morphism g : B → C, the Cmodule M , regarded as a B module via restriction of scalars along g, is Taylor reducedsince M is Taylor reduced (Lemma 5), so to give a B module map ΩB/A →M is the same

thing as giving a B module map Ω†B/A → M in light of the universal property of Taylor

reduction (§2.4).

Lemma 25. Suppose f : A → B is a local map of local sheaves of rings on a space X

inducing an isomorphism A†x → B†x for some x ∈ X. Then for any map C → A of sheaves

of rings on X, the natural map ΩA/C,x → ΩB/C,x is an isomorphism.

Proof. By (4.1.7), Lemma 37, and (2.4.2), we have natural isomorphisms

ΩA/C,x = ΩAx/Cx= Ω

Ax/Cx= Ω

A†x/Cx

and similarly with A replaced by B. But the hypothesis certainly ensures that ΩA†x/Cx

→ΩB†x/Cx

is an isomorphism.

Theorem 26. Suppose f : A → B is a surjective local map of local sheaves of rings ona space X inducing an isomorphism A† → B† on Taylor reductions. Then for any map

C → A of sheaves of rings on X, the natural map Ω†A/C → Ω†B/C is an isomorphism.

Proof. By definition of the Frechet differentials and the Taylor reduction (§2.4), we havea commutative diagram with exact rows

0 // Ω[A/C

// ΩA/C//

Ω†A/C//

0

0 // Ω[B/C

// ΩB/C// Ω†B/C

// 0.

Since f is surjective, so is ΩA/C → ΩB/C , and we reduce to showing that for any local

section ω ∈ ΩA/C(U) with df(ω) ∈ Ω[B/C(U) we have ω ∈ Ω[

A/C(U). To say that df(ω) is

in Ω[B/C(U) means that tx(df(ωx)) is zero in ΩB/C,x for every x ∈ U . But A† → B† is an

isomorphism, hence so is its stalk A†x → B†x and the completion A†x → B†x of this stalk, so

Lemma 25 implies that ΩA/C,x → ΩB/C,x is an isomorphism, hence tx(df(ωx)) = 0 for all

x ∈ U implies tx(ωx) = 0 in ΩA/C,x for every x ∈ U , hence ω ∈ Ω[A/C(U).

22 W. D. GILLAM

Corollary 27. Let f : X → Y be a map of locally ringed spaces and let X† → X be the

Taylor reduction of X. Then there is a natural isomorphism Ω†X/Y = Ω†X†/Y

of sheaves

on X† (O†X modules).

Proof. Apply the theorem with C = f−1OY , A = OX , B = O†X .

4.3. Assumptions. We now begin to specialize our general theory of Frechet differentialsto the category OS of O-spaces. For X ∈ RS/k, set ΩX := ΩX/k. Write Ωn,0 for the On,0

module of Kahler differentials of k → On,0. For the remainder of this section, we assumethat the admissible algebra of functions O satisfies the following additional condition:

A4. The module Ωn,0 is freely generated by dx1, . . . , dxn.

The condition (A4) implies Ωkn is freely generated by dx1, . . . , dxn, hence the map

w : kn → Spec k[x1, . . . , xn]

of (A1) induces an isomorphism w∗ΩSpec k[x] → Ωkn . Since On,0 is reduced, hence Taylor

reduced, so is Ωkn∼= On

n, so we have Ω†kn = Ωkn .

4.4. Inverse limits. We now prove that Frechet differentials commute with finite inverselimits in OS.

Theorem 28. Let (fi : Xi → Yi)i be a finite inverse limit system in OS with inverse limitf : X → Y and projections pi : X → Xi, qi : Y → Yi. Then the natural map (4.1.11) isan isomorphism.

Proof. It suffices to treat the cases of products of two factors and equalizers. For productsof two factors, consider maps fi : Xi → Yi (i = 1, 2). We must show that

π∗1ΩX1/Y1⊕ π∗2ΩX2/Y2

→ ΩX/Y

is an isomorphism. This is local (the construction of products of O spaces is local), so wecan assume each O-space involved is a model:

Xi = Z(Ii) ⊆ Ui ⊆ (kmi ,Omi)

Yi = Z(Ji) ⊆ Vi ⊆ (kni ,Omi)

X = Z((p−11 I1 + p−12 I2)T ) ⊆ U1 × U2 ⊆ (km,Om) (m := m1 +m2)

Y = Z((q−11 J1 + q−12 J2)T ) ⊆ V1 × V2 ⊆ (kn,On) (n := n1 + n2).

4.5. O-ification.

Theorem 29. Let X be a scheme of locally finite type over k, τ : XOS → X the associatedO-space (3.10). The natural map τ∗ΩX → ΩXOS is an isomorphism.

Proof. This is local, so we can assume X = Spec k[x1, . . . , xn]/(f1, . . . , fm).


Appendix: Completion

Let A be a ring, I ⊆ A an ideal. The I-adic topology on A is the unique topology on Awhere the translations a 7→ a+b are continuous for every b ∈ A and where In : n ∈ N is abasis for the open neighborhoods of 0. Equivalently, it is the smallest topology making theprojection A→ A/In continuous for every n when A/In is given the discrete topology, or,the smallest topology making the product of these projections A→

∏nA/I

n continuous.The I-adic topology makes A a topological ring in the sense that addition, subtraction,and multiplication are continuous maps A × A → A. If B is another ring with an idealJ ⊆ B, then a ring map f : A→ B is continuous for the corresponding adic topologies iffIn ⊆ f−1(J) for some n ∈ N. In particular, a local map of local rings is a continuous mapfor the m-adic topologies.

Let A be a topological ring. We say that A is separated iff the intersection of allneighborhoods of 0 is 0. Equivalently, A is separated iff the underlying topologicalspace is T1 in the usual sense [Eng, p. 36]. A sequence a1, a2, . . . of elements of A is calledCauchy iff, for any neighborhood U of 0, there is an N ∈ N such that am − an ∈ U forall m,n ≥ N . An element a ∈ A is called a limit of the sequence a1, a2, . . . iff, for anyneighborhood U of 0, there is an N ∈ N such that a − an ∈ U for all n ≥ N . If A isseparated then any sequence has at most one limit. The converse holds if the topology onA is first countable in the sense that there is a countable basis for neighborhoods of 0. Inparticular, the I-adic topology on A is separated iff every sequence in A has at most onelimit. A topological ring is called complete iff every Cauchy sequence has a unique limit.One can construct the completion of a topological ring in great generality by consideringequivalence classes of Cauchy sequences in the usual fashion.

We will only be interested in the case of the I-adic topology on a ring A, in which casethe I-adic completion of A is given by

A := lim←−

A/In.

The ring A is of course topologized as a subspace of the product of the A/In (these are

given the discrete topology), hence it is clear the natural map t : A → A is continuous

with dense image. It is straightforward to check that A is complete and separated, that

the ideal of A generated by t(I) coincides with the completion I = lim←−

I/In, and that the

topology on A defined previously coincides with the I-adic topology. If f : A → B is aring map with f(In) ⊆ J for an ideal J ⊆ B, then we have a commutative diagram ofcontinuous maps of topological rings as below.

Af //

t

B

t

Af // B

In the case of the m-adic topology on a local ring A, the completion A is also a localring with maximal ideal m. The following useful lemma is standard and straightforwardto prove.

Lemma 30. Let f, g : A→ B be two continuous maps of topological rings which agree ona dense subset X ⊆ A. Suppose that B is separated. Then f = g.

24 W. D. GILLAM

The main purpose of this appendix is to establish some basic properties of the I-adiccompletion functor

Mod(A) → Mod(A)

M 7→ M := lim←−

M/InM.

4.6. Completion map. There is a natural map t : M → M which is t : A → A linear.Since kernels commute with inverse limits, we have

Ker t = lim←−

Ker(M → InM)(4.6.1)

= ∩nInM.

Lemma 31. Suppose M ′ ⊆ Ker(t : M → M). Then

(M/M ′)/(In(M/M ′)) = M/InM,

hence the natural map M → M/M ′ is an isomorphism.

Proof. It is easy to see that

(M/M ′)/(In(M/M ′)) = M/(M ′ + InM),

but M ′ ⊆ Ker t ⊆ InM , so (M ′ + InM) = InM , which proves the first statement; thesecond statement then follows by taking inverse limits.

4.7. Exactness. In general, the completion functor is neither left exact nor right exact.This is because it is obtained as the composition of the functors

Mod(A) → Mod(A)N(4.7.1)

M 7→ (n 7→M/InM = M ⊗A (A/In))

(regard N as a category with a unique morphism m→ n iff m ≥ n) and

lim←−

: Mod(A)N → Mod(A),

the first of which is right but not left exact, and the second of which is left but not rightexact. Completion does, however, commute with finite direct products (= finite directsums) since this is true of tensor products and inverse limits.

It is perhaps surprising that, when A is noetherian, so is A, and the completion functoris both left and right exact when restricted to finitely generated (equivalently, finitelypresented) A modules. It follow formally from the exactness of completion that the natural

map M ⊗A A→ M is an isomorphism of A modules for any finitely generated M (applythe Five Lemma to a presentation of M using right exactness of the tensor product). Wewill establish a somewhat more subtle variant of this in Corollary 36 in the non-noetheriansetting.

In this article, we almost never deal with noetherian A, so we need some exactnessresults that do not require noetherian hypotheses.

Lemma 32. If 0 → M ′ → M → M ′′ → 0 is an exact sequence of A modules with M ′′

flat, then 0→ M ′ → M → M ′′ → 0 is exact.


Proof. By the hypothesis on M ′′, the image of this exact sequence under (4.7.1) remainsexact; the image of the latter under lim

←−is then exact because the transition functions for

the inverse limit system n 7→ M ′/InM ′ are surjective, so it trivially satisfies the Mittag-Leffler condition.

Lemma 33. For any n ∈ N and any A module M we have natural isomorphisms

M/InM = M/InM = M/InM.

Proof. The point is that

(M/In)/(Im(M/InM)) = M/(ImM + InM),

which is just M/InM as long as m ≥ n, for then we have ImM ⊆ InM , so taking inverselimits over m gives the first isomorphism. When m ≥ n, we have an exact sequence

0→ InM/ImM →M/ImM →M/InM → 0.

As an inverse limit system in m, the one on the left has surjective transition functions, sothe sequence stays exact when we take inverse limits over m. Note

lim←−

InM/ImM = lim←−

InM/(Im−n(InM))

= lim←−

InM/(Im(InM))

= InM.

4.8. Remark. The same argument is used to prove exactness of completion in the noe-therian case. When the bottom row of

0 // (InM ∩M ′) _

// InM _

// InM ′′ _

// 0

0 // M ′ // M // M ′′ // 0

is exact, it is easy to see that the top row is also exact, so the sequence of quotients

0→M ′/(InM ∩M ′)→M/InM →M ′′/InM ′′ → 0

is also exact by the Snake Lemma. The transition functions for the inverse limit systemn 7→M ′/(InM ∩M ′) are surjective, so if we set

M′

:= lim←−

M ′/(InM ∩M ′)

then Mittag-Leffler says

0→M′ → M → M ′′ → 0

is exact. In particular, the completion of a surjection f : M → M ′′ is again a surjection.

The difficulty is to show that the natural map M′ → M ′ is an isomorphism. This is done

by appeal to Artin-Rees: When A is noetherian and M is finitely generated, the filtrationsI•M ∩M ′ and I•M ′ on M ′ are commensurate.

26 W. D. GILLAM

4.9. Exactness results. Here we prove that m-adic completion is right exact on finitely

generated modules over a local ring (A,m) when t : A → A is surjective and A is noe-therian. The main example we have in mind is when A = C∞0 is the ring of germs of C∞functions on Rn at the origin and I is its unique maximal ideal:

I = f ∈ C∞0 : f(0) = 0= 〈x1, . . . , xn〉.

In this case, the completion map t is given by the Taylor series

t : C∞0 → R[[x]]

f 7→∑

(e1,...,en)∈Nn

∂e1

∂xe11· · · ∂

en

∂xe1nf

∣∣∣∣0

xe11 · · ·xenne1! · · · en!

and is surjective by a theorem of Borel [Mal, I.4].

In the rest of this section, we always consider a local ring (A,m). Completion M 7→ Mis always m-adic completion.

Lemma 34. Suppose 0 → M ′ → M → M ′′ → 0 is an exact sequence of modules over a

local ring (A,m). Assume A is noetherian, M is finitely generated (as an A module), and

t : M → M is surjective. Then the sequences

0→ t(M ′)→ M → M ′′ → 0

M ′ → M → M ′′ → 0

are exact sequences of A modules. In the notation of Remark 4.8, t(M ′) = M′.

Proof. First we claim that

Ker(M → M ′′/mnM ′′) = t(M ′) + mnM.

The containment ⊇ is obvious. Suppose m ∈ M is in this kernel. Since t is surjective,we can find m ∈ M such that t(m) = m. Using Lemma 33 and commutativity, we seethat m maps into mnM ′′ ⊆ M ′′ via the map f : M → M ′′, so we can write f(m) = im′′

for i ∈ mn,m′′ ∈ M ′′. Choose m ∈ M so that f(m) = m′′. Then f(m − im) = 0, som− im ∈M ′ and we have

m = t(m)

= t(m− im) + it(m)

∈ t(M ′) + mnM,

which establishes the other containment.

It follows from Lemma 33 that M ′′ is the inverse limit of the M ′′/mnM ′′ = M ′′/mnM ′′,

so, since inverse limits commute with kernels, the kernel of f : M → M ′′ is the inverselimit of the kernels discussed above, so

Ker f = ∩n(t(M ′) + mnM)

= t(M ′).

For the last equality, we need to argue that K := ∩nmnM is zero in M/t(M ′). Since

M is finitely generated over the noetherian ring A, so are M/t(M ′) and K, so by the


Krull Intersection Theorem we have mK = K, hence K = 0 by Nakayama’s Lemma. Thisestablishes exactness in the one place where it is not obvious.

Theorem 35. Suppose (A,m) is a local ring, A is noetherian, and t : A→ A is surjective.Then for any exact sequence 0 → M ′ → M → M ′′ → 0 of A modules with M finitely

generated, the sequence M ′ → M → M ′′ → 0 is exact. In particular, completion is rightexact on the category of finitely generated A modules.

Proof. Since M is finitely generated, we can find an exact sequence

0→ K → An →M → 0(4.9.1)

for some finite n. Since t : A → A is surjective, so is t : An → An = An (completion

commutes with finite products/sums). Applying Lemma 34 to (4.9.1), we see that An → M

is surjective (so M is a finitely generated A module) and t : M → M is surjective. Thetheorem now follows immediately from Lemma 34.

Corollary 36. Suppose (A,m) is a local ring, A is noetherian, and t : A→ A is surjective.

The completion map M 7→ M takes finitely generated A modules to finitely presented A

modules and the natural map M ⊗A A → M is surjective for any finitely generated Amodule M , and an isomorphism for any finitely presented A module M .

Proof. When M is finitely generated we have an exact sequence 0→ K → An → M → 0of A modules. By the theorem and right exactness of the tensor product, we have acommutative diagram with exact rows

K ⊗A A //

An ⊗A A //

∼=

M ⊗A A

// 0

K // An // M // 0,

where the indicated arrow is an isomorphism because tensor products and completion

both commute with finite direct sums (= finite direct products). In particular, M isfinitely generated.8 The right vertical arrow is surjective by an easy diagram chase. If,furthermore, M is finitely presented, then we may take K finitely generated, hence theleft vertical arrow is surjective by what we just proved, hence the right vertical arrow isan isomorphism by a slightly harder diagram chase (Subtle Five Lemma).

4.10. Differentials. Here we extract some results concerning Kahler differentials andcompletions from our study above.

Lemma 37. Suppose A is ring, I ⊆ A is an ideal, A is the I-adic completion of A andf : C → A is an arbitrary ring homomorphism. Then there is a natural isomorphism

ΩA/C

= ΩA/C of A modules.

Proof. Let t : A→ A denote the completion map. The Leibnitz Rule implies that

d(In+1) ⊆ InΩA/C(4.10.1)

8We already saw this in the proof of the theorem. Note that finitely generated/presented are the same

thing because we assume A is noetherian.

28 W. D. GILLAM

for all n ∈ N. Consider a typical element

a = (a1, a2, . . . ) ∈ A = lim←−

A/In.

Here an ∈ A/In and we have an+1 − an ∈ In for some (equivalently any) lifts an ∈ A ofthe an. In light of (4.10.1), the image of dan+1 ∈ ΩA/C in ΩA/C/I

nΩA/C is independent of

the choice of lift an+1 ∈ A of an+1 ∈ A/In+1, hence we may write dan+1 ∈ ΩA/C/InΩA/C

unambiguously. Formula (4.10.1) also implies that

Da := (da2, da3, . . . ) ∈∞∏n=1

ΩA/C/InΩA/C

in fact lies in ΩC/A because, for n = 1, 2, . . . , we have

dan+2 − dan+1 = d(an+2 − an+1) ∈ InΩA/C

since an+2 − an+1 ∈ In+1. The desired natural isomorphism ΩA/C

→ ΩA/C is given by

da 7→ Da. It is straightforward to check that this is well-defined—i.e. it kills dtf(c) ∈ ΩA/C

for every c ∈ C and it kills the Leibnitz relation. The inverse is constructed as follows.

The completion map t induces a t linear map dt : ΩA/C → ΩA/C

. Recall that I = t(I)A,

so dt(InΩA/C) ⊆ InΩA/C

. Given an element

ω = (ω1, ω2, . . . ) ∈ ΩA/C = lim←−

ΩA/C/InΩA/C ,

we hence have well-defined elements dt(ωn) ∈ ΩA/C

/InΩA/C

defined by choosing a lift

ωn ∈ ΩA/C of ωn ∈ ΩA/C/InΩA/C , then noting that the image of dt(ωn) ∈ Ω

A/Cin

ΩA/C

/InΩA/C

is independent of the chosen lift. The desired inverse is obtained by mapping

ω to

(dt(ω1), dt(ω2), . . . ) ∈ ΩA/C = lim←−

ΩA/C/InΩA/C .

Lemma 38. Suppose (A,m) is a local ring, A is noetherian, t : A→ A is surjective, andC → A is a ring homomorphism such that ΩA/C is a finitely presented A module. Then

the natural maps ΩA/C ⊗A A → ΩA/C and ΩA/C ⊗A A → ΩA/C

are isomorphisms of A

modules.

Proof. The map ΩA/C ⊗A A → ΩA/C is an isomorphism by Corollary 36 and hence theother map in question is also an isomorphism by the previous lemma.

Lemma 39. Suppose f : A → B is a local map of local rings such that f is an isomor-

phism. Then for any ring homomorphism C → A, the natural map ΩA/C → ΩB/C is an

isomorphism. Suppose furthermore that f is surjective, A is noetherian, t : A → A is

surjective, and ΩB/C finitely presented. Then the natural map ΩA/C ⊗A A→ ΩB/C ⊗B Bis an isomorphism.

Proof. Since f is an isomorphism, ΩA/C→ Ω

B/Cis certainly an isomorphism. But this

isomorphism is identified with ΩA/C → ΩB/C by Lemma 37, so the first statement is clear.


Under the additional hypotheses, we first note that surjectivity of f , f , and t : A → A

implies surjectivity of t : B → B. We then have a commutative diagram

ΩA/C

// ΩB/C

ΩA/C ⊗A A

OO

// ΩB/C ⊗B B

OO

where the vertical arrows are isomorphisms by the previous lemma and the top horizontal

arrow is an isomorphism since f is an isomorphism, hence the bottom horizontal arrow isalso an isomorphism.

References

[DS] J. A. Navarro Gonzalez and J. B. Sancho de Salas, C∞-differentiable spaces. Lec. Notes Math.1824. Springer-Verlag, 2003.

[EGA] A. Grothendieck and J. Dieudonne, Elements de geometrie algebrique. Pub. Math. I.H.E.S., 1960.[Eng] R. Engelking, General Topology. Heldermann Verlag Berlin, 1989.[GAGA] J. P. Serre, Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier (1956) Tome 6, 1–42.[G] W. D. Gillam, Localization of ringed spaces. Adv. Pure Math. (2011) 1(5), 250–263.[H] R. Hartshorne, Algebraic Geometry. Springer-Verlag, 1977.[Ill] L. Illusie, Complexe cotangent et deformations, I. Lec. Notes Math. 239. Springer-Verlag, 1971.[KU] K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures. Ann. Math.

Studies 169. Princeton Univ. Press, 2009.[Mal] B. Malgrange, Ideals of differentiable functions. Tata Inst. Fund. Res. Stud. Math. 3. Oxford

Univ. Press, 1967.[Mat] H. Matsumura, Commutative ring theory. Cambridge Stud. Adv. Math. 8. Cambrige Univ. Press,

1986.[Nav] J. A. Navarro, Differential spaces of finite type. Math. Proc. Camb. Phil. Soc. (1995) 117, 371-384.[Whi] H. Whitney, On ideals of differentiable functions. Amer. J. Math. 70 (1948), 635–658.

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