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ALGEBRAIZATION AND TANNAKA DUALITY

BHARGAV BHATT

1. INTRODUCTION

Our goal in this paper is to identify certain naturally occurring colimits of schemes and algebraic spaces.The statements, which are essentially algebraization results for maps between schemes and algebraic spaces,are elementary and explicit. However, our techniques are indirect: we use (and prove) some new Tannakaduality theorems for maps of algebraic spaces. Our approachto these theorems relies on a systematic deploy-ment of perfect complexes (ergo, we use some derived algebraic geometry) instead of ample line bundles orvector bundles. Consequently, the Tannaka duality resultswe obtain have fewer, and much weaker, finite-ness constraints than some of the existing ones: we only insist that our algebraic spaces be quasi-compactand quasi-separated (qcqs), and do not require any quasi-projectivity or noetherian hypotheses. All rings areassumed to be commutative.

1.1. Algebraization of jets. The first colimit we identify is that of an affine (adic) formalscheme.

Theorem 1.1. If A is a ring which isI-adically complete for some idealI, andX is a qcqs algebraic space,thenX(A) ≃ limX(A/In) via the natural map.

An equivalent formulation is: ifA = limA/In, thenSpec(A) is a colimit of the diagram{Spec(A/In)}in the category of qcqs algebraic spaces. Theorem1.1is straightforward to prove ifA/I is local; its contentbecomes apparent only whenSpec(A/I) has some non-trivial global geometry. Note also that there are nonoetherian assumptions on any object in sight, so the idealI might not be finitely generated. In fact, theresult extends to more general topological ringsA that arise naturally inp-adic geometry (see Remark4.3).This answers a question asked by Drinfeld, and has the following representability consequence in the theoryof arc spaces, which was our original motivation for pursuing Theorem1.1.

Corollary 1.2. If X is a qcqs algebraic space, then the “formal arc” functorArcsX(R) := X(RJtK) is anfpqc sheaf on the category of rings, and is identified with thefunctorR 7→ limX(R[t]/(tn)).

As the functorArcsX is almost never locally finitely presented (even forX an algebraic variety), onecannot reduce Corollary1.2 to the corresponding assertion on the category of noetherian rings (which iseasier to prove). This corollary answers a question raised in [NS10, §2] and pointed out to us by Nicaise.The following feature of the proof of Theorem1.1 seems noteworthy: given a compatible system{ǫn :Spec(A/In) → X} ∈ limX(A/In), we construct an algebraizationǫ : Spec(A) → X without evermusing about points ofSpec(A) \ Spec(A/I).

1.2. Algebraization of products. The second result deals with products, rather than cofiltered inverse lim-its, of rings; this question was brought to our attention by Poonen.

Theorem 1.3. If {Ai}i∈I is a set of rings, andX is a qcqs algebraic space, thenX(∏

iAi) ≃∏

iX(Ai)via the natural map.

An equivalent formulation is: the schemeSpec(∏

iAi) is a coproduct of{Spec(Ai)} in the category ofqcqs algebraic spaces. Note thatsomefiniteness hypothesis onX is necessary: the (typically non-quasi-compact) scheme⊔iSpec(Ai) is a coproduct of{Spec(Ai)} in the category of all schemes. The non-trivialcase, again, is when the ringsAi have interesting global geometry. Moreover, like Theorem1.1, the proof of

1

Theorem1.3also circumvents ever contemplating points ofSpec(∏

iAi) \ ⊔iSpec(Ai). Theorem1.1maybe used to describe adelic points on algebraic spaces over global fields (see Corollary8.7).

1.3. Formal glueing. The third result concerns the classical Beauville-Laszlo theorem [BL95], which isincredibly useful in the construction of bundles on families of curves arising, for example, in geometricrepresentation theory. Recall that this theorem asserts: given an affine schemeX with a Cartier divisorZ ⊂X, one can patch compatible quasi-coherentsheaveson X (the completion ofX alongZ) andU := X \Zto a quasi-coherent sheaf onX, provided the sheaves being patched are flat alongZ. This is a sheaf-theoreticmanifestation of the principle thatX is an algebro-geometric analogue of a tubular neighbourhood of Z inX, soX behaves as though it were built by glueingX to U overX \ Z = X ×X U . In the next theorem,we vivify this geometric intuition by showing thatX is literally the pushout ofX andU alongX ×X U ,which perhaps clarifies the glueing result for sheaves. Along the way, we also offer an improvement on theglueing result itself: the patching works unconditionallyfor quasi-coherentcomplexes.

Theorem 1.4. Letπ : Y → X be a map of qcqs algebraic spaces. Assume there exists a finitely presentedclosed subspaceZ ⊂ X satisfying1 Z ×L

X Y ≃ Z. SetU = X \ Z andV = Y \ π−1(Z). Then:

(1) The fibre square

Vj

//

π��

Y

π��

Uj

// X

is a pushout in qcqs algebraic spaces.(2) The natural map induces an equivalenceD(X) ≃ D(Y )×D(V ) D(U).

HereD(X) is the∞-category of quasi-coherent complexes onX. The Beauville-Laszlo theorem con-cerns the special case whereX = Spec(A) for some ringA, Z = Spec(A/f) for f ∈ A a regular element,andY = Spec(limA/fn) is the completion. In this case, they show an analogue of (2) for modules thataref -regular. In order to get the general consequence (2) above,it is crucial to work with∞-categories: thecorresponding statement about the full module category or the classical derived category is false.

1.4. Tannaka duality. The “surjectivity” assertions in Theorem1.1 and Theorem1.3, as well as (2) inTheorem1.4, may be viewed as algebraization results for maps. Despite the elementary formulations, wedo not have a constructive proof of any of these, even for schemes, except in special cases. Instead, thedesired algebraization is constructed by first building a suitable functor on (derived) categories of quasi-coherent sheaves; Tannaka duality results then show that this functor is the pullback functor for a morphism.

The implementation of the strategy above necessitates certain derivedTannaka duality results.2 Theseduality results rely crucially on Lurie’s [Lur11d], but cannot directly be deduced from it: Lurie works ingreater generality, and consequently has stronger hypotheses. Nevertheless, leveraging his ideas with somemore classical techniques, we show the following, which suffices for the applications above.

Theorem 1.5. If X andS are algebraic spaces withX qcqs, then pullback induces equivalences

Hom(S,X) ≃ Fun⊗(Dperf(X),Dperf (S)) ≃ FunL⊗(D(X),D(S)).

HereDperf(X) ⊂ D(X) is the full subcategory of perfect complexes onX, and similarly forS. Also,Fun⊗(Dperf(X),Dperf (S)) parametrizes exact symmetric monoidal functorsDperf(X) → Dperf(S), whileFunL⊗(D(X),D(S)) parametrizes cocontinuous (i.e., colimit-preserving) symmetric monoidal functors.Note that, due to the existence of Fourier-Mukai transforms, there is no hope of proving such a result withoutkeeping track of the⊗-structure. The relevance of Theorem1.5 to the previous discussion on colimits is:

1The conditionZ ×LX Y ≃ Z means: the mapsY → X andZ → X are mutuallyTor-independent, andπ−1(Z) ≃ Z.

2It is easy to see why the derived setting is preferable in approaching Theorem1.1: perfect complexes are easier to manipulatethan (finitely presented) quasi-coherent sheaves, especially with respect to operations involving both limits and tensor products.

2

Corollary 1.6. Fix a qcqs algebraic spaceX, and a diagram{Xi} of qcqs spaces overX. If pullbackinducesDperf(X) ≃ limDperf(Xi), thenX ≃ colimXi in the category of qcqs algebraic spaces.

Besides the applications above, Corollary1.6should be also useful in excising hypotheses on the diagonalin certain existence results; for example, we indicate in Remark 4.5 why the separatedness assumption inGrothendieck’s formal geometry version of Chow’s theorem can be dropped completely.

As mentioned above, Lurie proved a related Tannakian resultfor a very general class of (spectral) derivedstacks in [Lur11d, Theorem 3.4.2]. When specialized to algebraic spaces, hisresult differs from Theorem1.5 in two ways: he requires the diagonal ofX to be affine, and he “only” shows that cocontinuous sym-metric monoidal functorsF : D(X) → D(S) that preserve connective objects andflat objectscome fromgeometry. The first restriction is relatively mild, at leastin applications, but the last one is severe, renderinghis result inapplicable to Theorems1.1, 1.3, and1.4 (as it is quite difficult to control flatness propertiesof modules through limits). An analogue of Theorem1.5 for noetherian stacks with some tameness andquasi-projectivity hypotheses (over a field) can be found in[FI13]. A generalization of Theorem1.5 to afairly large class of stacks, together with some applications, is also the subject of forthcoming joint work ofDaniel Halpern-Leistner and the author.

In the world of schemes, one can go further than Theorem1.5to get an underived statement. In fact, Luriealready did so [Lur04] for algebraic stacks under the afore-mentioned constraints, and these were removedby Brandenburg and Chirvasitu in the case of schemes to show:

Theorem 1.7. [BC12] If X andS are schemes withX qcqs, then pullback induces an equivalence

Hom(S,X) ≃ FunL⊗(QCoh(X),QCoh(S)).

HereFunL⊗(QCoh(X),QCoh(S)) denotes the category of all cocontinuous symmetric monoidal functorsbetween the abelian categories of quasi-coherent sheaves on X andS. We can use this result, in lieu ofTheorem1.5, to prove Theorem1.3 in the world of schemes. We conclude this introduction by recording astrengthening of Theorem1.7in a special case that arises often in practice.

Proposition 1.8. Fix schemesX andS. If X is qcqs with enough vector bundles, then pullback induces

Hom(S,X) ≃ FunL⊗(Vect(X),Vect(S)).

Here the assumption onX means that every finitely presented quasi-coherent sheaf isthe cokernel of amap of vector bundles; any scheme that is quasi-projective over an affine has this property, and this propertyis studied in depth in [Tot04]. The objectFunL⊗(Vect(X),Vect(S)) denotes the category of right exactsymmetric monoidal functorsVect(X) → Vect(S). It is important to note that the property of being “rightexact” for a sequence of bundles is not intrinsic to the category of vector bundles: one needs the ambientcategory of all quasi-coherent sheaves to make sense of it. The quasi-projective case of Proposition1.8wasalso shown much earlier by Savin [Sav06] with a different proof.

Sketch of proofs. We begin with Theorem1.1. The injectivity ofX(A) → limX(A/In) is relatively ele-mentary. For surjectivity, given compatible maps{ǫn : Spec(A/In) → X}n∈N, one must construct a mapǫ : Spec(A) → X algebraizing{ǫn}. If X is a quasi-projective variety, thenX has “enough” vector bundles:every quasi-coherent sheaf can be “approximated” by finite complexes of vector bundles. The data{ǫn} de-fines a functorF : Vect(X) → Vect(Spec(A)) as the composition of the pullbacklim ǫ∗n : Vect(X) →limVect(Spec(A/In)) and the inverse of the equivalenceVect(Spec(A)) ≃ limVect(Spec(A/In)) (seeLemma4.11). One then checks thatF preserves exact sequences, so Proposition1.8 gives the desiredmapǫ : Spec(A) → X. In general,X might not admit a single non-trivial vector bundle, which rendersthis approach useless. However,X always has enough perfect complexes by a fundamental resultgoingback to Thomason [TT90]. Hence, the preceding strategy can be salvaged at the derived level using perfectcomplexes, instead of vector bundles, and Theorem1.5.

The proof of Theorem1.3is similar, but the construction of an appropriate pullbackfunctorDperf(X) →Dperf(Spec(

∏iAi)) associated to a family of mapsǫi : Spec(Ai) → X is harder: one must show that if

3

K ∈ Dperf(X), then∏

i f∗i K is a perfect(

∏iAi)-complex. We check this by verifying that the “size”

of f∗i K is bounded independently ofi, which, in turn, is accomplished via an analysis of the number ofsections needed to generate a module over a ring once the corresponding numbers over a Nisnevich coverhave been specified. More details can be found at the start of§7.

For Theorem1.5, the full faithfulness ofHom(S,X) → FunL⊗(D(X),D(S)) is a consequence of a resultof Lurie. For essential surjectivity, fix a cocontinuous symmetric monoidal functorF : D(X) → D(S). Wefirst check thatF preserves connecitivty; this allows us to “pull back” affineX-spaces to affineS-spacessimply by applyingF to the corresponding commutative algebra inD≤0(X). Viewing a quasi-affineX-space as the complement of a (constructible) closed subspace in an affineX-space, one may also pull backquasi-affineX-spaces alongF . The crucial assertion is that this procedure respects etale morphisms as wellas coverings; this is deduced by showing the analogous assertions for arbitrary commutative algebras inD(X) as pushing forward the structure sheaf gives a fully faithful embedding of quasi-affineX-spaces intocommutative algebras inD(X). Consequently, the construction off : S → X such thatf∗ = F is etalelocal onX, so we reduce to the case whereX is affine, which is easy.

Proposition1.8 is deduced painlessly from Theorem1.7 by writing quasi-coherent sheaves as filteredcolimits of cokernels of maps of vector bundles. The key observation is that one can recoverQCoh(X)from Vect(X) equipped with the (extra) data of the class of surjective maps.

For Theorem1.4, we first check that pullback inducesDZ(X) ≃ Dπ−1(Z)(X)3. The rest of the proof of(1) is then a formalisation of the idea thatD(X) is an extension ofD(U) byDZ(X), and thatD(Y ) is anextension ofD(V ) byDπ−1(Z)(Y ). Finally, Theorem1.5 immediately yields (2) from (1).

An outline of the paper. We begin by proving Theorem1.5 in §2; this section contains the most seriousdose of derived algebraic geometry in this paper, and one canfind outsider-friendly discussions of some keynotions in [Gro10], [Toe14], and [BZFN10, §2–3]. The non-derived analogue of Theorem1.5 for schemes(i.e., Brandenburg and Chirvasitu’s Theorem1.7, as well as Proposition1.8) is the subject of§3. Theorem1.1 is then taken up in§4; we also discuss examples illustrating the limit of such results. Formal glueingresults, including Theorem1.4, are the focus of§5, though we begin by establishing Corollary1.6 usingTheorem1.5. Theorem1.3is proven across§6 and§7: the former contains a non-derived proof for schemesusing Theorem1.7, while the latter handles algebraic spaces using Theorem1.5 (and is independent of theformer). Finally, the limits of Theorem1.3are explored through some examples in§8.

Notation. We use the language of∞-categories from [Lur09], except that we use the term “(co)continuousfunctor” to describe (co)limit preserving functors. For an∞-categoryC, we writeh(C) for its homotopy-category. For a mapf : K → L in a stable∞-category, we writefib(f) andcofib(f) for the fibre andcofibre respectively. A functor between stable∞-categories is always assumed to be exact.

For a symmetric monoidal∞-categoryC, we writeCAlg(C) for the∞-category of commutative algebraobjects (in the sense ofE∞-rings; see [Lur11b, §2.1]). If C is an ordinary category, so isCAlg(C): thelatter coincides with the classical defined category of commutative monoids inC (by [Lur11b, Example2.1.3.3]). We will often use the notion of a dualizable object in a symmetric monoidal∞-category and itsbasic properties (see [Lur11b, §4.2.5]). In particular, we will freely use that such objectsare preserved bysymmetric monoidal functors, and thatK ⊗− is continuous ifK is dualizable.

For symmetric monoidal∞-categoriesC andD, let Fun⊗(C,D) denote the∞-category of symmetricmonoidal functorsC → D. We use a superscript of “L” to denote the class of cocontinuous functors, whilea subscript of “c” denotes the class of functors preserving compact objects.For example,FunL⊗,c(C,D) isthe full subcategory ofFun⊗(C,D) spanned by cocontinuous symmetric monoidal functorsF : C → D that

3HereDZ(X) ⊂ D(X) denotes the full subcategory of complexes acyclic onX −Z, i.e., the kernel ofD(X) → D(X −Z).In particular, the inclusionDZ(X) ⊂ D(X) has a right adjointΓZ(−) : D(X) → DZ(X) given by Grothendieck’s theory oflocal cohomology. We prefer working with local cohomology,instead of completions, to access geometry “near”Z in Theorem1.4as the former has better compatibility properties with geometric operations onD(X), such as pullback and pushforward.

4

preserve compact objects; these notions are typically usedonly whenC andD are presentable with enoughcompact objects.

For any algebraic spaceX, let D(X) be the quasi-coherent derived category ofX (viewed as a sym-metric monoidal stable∞-category; see [Lur11d, Definition 2.3.6]), and letDperf(X) ⊂ D(X) be the fullsubcategory of perfect complexes. For qcqsX, we will repeatedly use:D(X) is a compactly generatedstable symmetric monoidal∞-category, andDperf(X) ⊂ D(X) coincides simultaneously with the classof compact objects and the class of dualizable objects; see [BvdB03, §3.3], [LN07, §4], [Sta14, Tag 09IU],[Lur11d, Corollary 2.7.33], and [Lur11c, Corollary 1.5.12]. We will also use a version of this with sup-ports: if Z ⊂ X is a constructible closed subspace, thenDZ(X) is compactly generated with compactobjects given byDZ(X) ∩Dperf(X), soDZ(X) ⊂ D(X) is the smallest full stable∞-subcategory closedunder colimits that containsDZ(X) ∩ Dperf(X); see [Rou08, Theorem 6.8] and [SS03, Lemma 2.2.1].All operations involving these objects are assumed to take place in the appropriate∞-categorical sense;for example, if{Xi} is a diagram of algebraic spaces, thenlimD(Xi) is the∞-categorical limit in thesense of [Lur09, §4]. Let QCoh(X) := D(X)♥ be the abelian category of quasi-coherent sheaves, andDcl(X) := h(D(X)), i.e., the classical derived category of complexes ofOX -modules with quasi-coherentcohomology sheaves. LetAff/X be the∞-category of affine morphisms overX (in the world of spectralalgebraic spaces, soAff/X ≃ CAlg(D≤0(X))), and letQAff/X be∞-category of quasi-affine morphismsoverX. As a rule, all geometric functors (suchf∗, f∗, Γ(X,−), ΓZ(−), etc.) are assumed to be derived,except in§3 and§6; we will use adornments (such as⊗L

A instead of⊗A) if there is potential for confusion.In §2, we will encounter some potentially non-connective commutative algebrasA ∈ CAlg(D(X)) forX

a qcqs algebraic space. The associated symmetric monoidal∞-categoryD(X,A) of A-modules inD(X)

(denotedModCommA (D(X))⊗ in [Lur11b, §3.3]) plays an important role, so we give a relatively concrete

(but imprecise) description. IfX = Spec(R) is affine, thenD(X) ≃ D(R) via Γ(X,−), soA definesa commutativeR-algebraA := Γ(X,A), andD(X,A) ≃ D(A) via Γ(X,−) (see [Lur11b, Corollary3.4.1.9]), where we writeD(A) for the∞-category ofA-module spectra (denotedModA in [Lur11b, §8]).In general, if we writeX = colimUi as a colimit of a diagram{Ui} of affine schemesUi = Spec(Ri) etaleoverX, thenD(X,A) ≃ limD(Ui,A|Ui

) ≃ limD(Ai) viaK 7→ {Γ(Ui,K|Ui)}, whereAi := Γ(Ui,A) is

the commutativeRi-algebra of global sections ofA|Ui. Moreover, the forgetful functorD(X,A) → D(X)

is conservative, continuous ([Lur11b, Corollary 4.2.3.3]), and cocontinuous ([Lur11b, Corollary 4.2.3.5]).In the special caseA = OX , this discussion recovers the etale descent equivalenceD(X) ≃ limD(Ui) ≃limD(Ri). An important non-connective example is given byA = j∗OU for j : U → X a quasi-affinemorphism; in this case,j∗ inducesD(U) ≃ D(X,A) (see [Lur11d, Corollary 2.5.16]), and the equivalenceD(X,A) ≃ limD(Ui,A|Ui

) above reduces to the etale descent equivalenceD(U) ≃ limD(U ×X Ui).For further psychological comfort, note that the Raynaud-Gruson devissage (see [RG71, Theorem 5.7.6],[Sta14, Tag 08GL] and [Lur11c, Theorem 1.3.8]) allows us to choose afinite diagram{Ui} realizingX.

Acknowledgements. I am very grateful to Vladimir Drinfeld, Johannes Nicaise, and Bjorn Poonen forbringing the algebraization questions treated here to my attention; to Johan de Jong and Ofer Gabber for en-lightening discussions, which had a conspicuous influence on this work; to Jacob Lurie, Bertrand Toen, andGabriele Vezzosi for conversations and communications that greatly improved my understanding of derivedalgebraic geometry, and consequently contributed indirectly, but significantly, to§2; to Daniel Halpern-Leistner and Brandon Levin for useful discussions; and especially to Brian Conrad: his numerous sugges-tions significantly improved the readability of this paper,and his insistence on the “correct” generality (inthe form of comments on an earlier note proving only Theorem1.1for schemes using [Toe12] and [BC12])led to Theorem1.5 in the first place. I was supported by NSF grants DMS 1340424 and DMS 1128155,

Remark 1.9. On circulation of the main results in this manuscript, we learnt that some of these were knownto some experts, at least under mild conditions: Lurie informed us that that Theorem1.5was familiar to himwhenX has affine diagonal, and Gabber has told us that he was roughlyaware of Theorem1.1and Theorem1.3through a potential extension of [BC12] to qcqs algebraic spaces.

5

2. TANNAKA DUALITY FOR ALGEBRAIC SPACES

The goal of this section is to prove the following:

Theorem 2.1. If X andS are qcqs algebraic spaces, then pullback induces isomorphisms

Hom(S,X) ≃ Fun⊗(Dperf(X),Dperf (S)) ≃ FunL⊗,c(D(X),D(S)) ≃ FunL⊗(D(X),D(S)).

Recall thatFun⊗(Dperf(X),Dperf (S)) is the∞-category of exact symmetric monoidal functorsDperf(X) →Dperf(S). We begin with the purely categorical aspects.

Lemma 2.2. There are natural identifications

Fun⊗(Dperf(X),Dperf (S)) ≃ FunL⊗,c(D(X),D(S)) ≃ FunL⊗(D(X),D(S)).

Proof. The first identification is a consequence ofD(X) = Ind(Dperf(X)). For the second, we must showthat every cocontinuous symmetric monoidal functorF : D(X) → D(S) preserves perfect complexes. AsDperf(X) ⊂ D(X) is the full subcategory of dualizable objects by [Lur11d, Corollary 2.7.33], this is animmediate consequence of symmetric monoidal functors preserving dualizable objects. �

The preceding identifications will be used without comment in the sequel. We now check full faithfulnessof Hom(S,X) → Fun⊗(Dperf(X),Dperf (S)).

Proof of full faithfulness.The functorsHom(−,X) andFun⊗(Dperf(X),−) are stacks for the Zariski (infact, fpqc), topology, so we may assumeS is affine. In this case, any mapS → X is quasi-affine. Thus, thefull faithfulness follows from Lurie’s theorem [Lur11d, Proposition 3.3.1]. �

For essential surjectivity, fix anF ∈ FunL⊗,c(D(X),D(S)). As before, we are free to localize onS, sowe assumeS is affine. We will useF to progressively build compatible etale hypercovers ofX andS by(quasi-)affine schemes. The first, and most essential, step is to “localize” algebraic geometry overS in termsof sheaf theory overX via a right adjoint toF ; if F arises from geometry, then this adjoint is simply thepushforward. The construction of this adjoint highlights the utility of using the functorF : D(X) → D(S),instead of its restriction to the full subcategories of perfect complexes.

Lemma 2.3. F admits a cocontinuous and conservative right adjointG : D(S) → D(X). Moreover,G is lax monoidal, and induces a symmetric monoidal equivalence D(S) ≃ D(X,GOS). Under thisequivalence, the functorF : D(X) → D(S) corresponds to the base change functorD(X) → D(X,GOS).

AsG is lax monoidal, the objectGOS is naturally anE∞-algebra, i.e., lifts canonically toCAlg(D(X));this explains the notationD(X,GOS).

Proof. The existence ofG follows from the adjoint functor theorem asF preserves colimits. The co-continuity ofG is a formal consequence ofF preserving compact objects (andD(X) being compactlygenerated). Moreover,Γ(S,−) onD(S) factors throughG by adjunction:Γ(S,K) ≃ Γ(X,GK). As Sis affine, it follows thatG is conservative. To get the monoidal behaviour, note that the right adjoint ofany symmetric monoidal functor is lax monoidal by [Lur11d, Proposition 3.2.1]. For the last assertion,we use Barr-Beck-Lurie. To apply this theorem, we must identify the monad resulting from the adjunc-tion asK 7→ GOS ⊗ K. By [Lur11b, Corollary 6.3.5.18], it is enough to check that the naturalmapGOS ⊗ E → G(F (E)) is an equivalence for anyE ∈ D(X). In fact, we may restrict toE ∈ Dperf(X)by cocontinuity. For suchE, one checks thatHom(K,−) applied to either side isH0(S,F (K∨) ⊗ F (E))for anyK ∈ Dperf(X); this proves the claim by Yoneda asDperf(X) generatesD(X) under colimits. Tosee that that this equivalence is symmetric monoidal, we must show that the natural map induces an isomor-phismG(K) ⊗GOS

G(L) ≃ G(K ⊗ L) for K,L ∈ D(S). By cocontinuity, asD(S) is generated undercolimits byOS , we may assumeK = L = OS , whence it is clear. �

Using this picture, we can “pullback” commutative algebrasand modules inD(X) in a tractable way:6

Lemma 2.4. F induces a cocontinuous functorCAlg(D(X)) → CAlg(D(S)) with right adjointG. ForanyA ∈ CAlg(D(X)), there is an induced cocontinuous symmetric monoidal functor FA : D(X,A) →D(S,F (A)) that preserves compact objects, and is compatible withF under the forgetful functor. For amapA → B in CAlg(D(X)), there is a canonical identificationFB(LB/A) ≃ LF (B)/F (A).

Proof. The first assertion comes from [Lur11d, Remark 3.2.2]. The equivalenceD(S) ≃ D(X,GOS)carriesF (A) to A ⊗ GOS by Lemma2.3. In particular, the desired functorD(X,A) → D(S,F (A)) issimply the base change functorD(X,A) → D(X,A⊗GOS). One then easily checks thatFA is cocontin-uous, symmetric monoidal, and compatible withF ; the preservation of compact objects is a consequence ofthe forgetful right adjoint preserving colimits. Finally,the claim about cotangent complexes is immediatefrom Lurie’s perspective [Lur11b, §8.3] on the functor of points of the cotangent complex in an arbitrarypresentable∞-category. More precisely, it follows from the base change formula [Lur11b, Proposition8.3.3.7]; see also [Lur12, Proposition 1.1.2] for a similar assertion. �

The next task is to show thatF preserves connective objects. For this, we recall a result on quasi-affinemaps in the derived setting. First, note that (opposite of) the categoryAff/X of affineX-spaces is identifiedwith CAlg(D≤0(X)) via pushforward of the structure sheaf. By abuse of notation, for anyY ∈ Aff/X ,we writeOY ∈ CAlg(D≤0(X)) for the corresponding algebra. In the derived setting, thisdiscussion toquasi-affine maps, thanks to a result of Lurie:

Lemma 2.5. Let f : U → X be a quasi-affine morphism. Thenf∗ induces a symmetric monoidal equiv-alenceD(U) ≃ D(X, f∗OU ). Moreover, the functorU 7→ f∗OU determines a fully faithful functorQAffopp

/X → CAlg(D(X)).

Proof. Almost everything can be found in [Lur11d, Proposition 3.2.5 and Lemma 3.2.8]. These referencesdo not explicitly state that the equivalenceD(U) ≃ D(X, f∗OU ) is symmetric monoidal, so we prove ithere. GivenK,L ∈ D(U), we must check thatφK,L : f∗K ⊗f∗OU

f∗L→ f∗(K ⊗OUL) is an equivalence.

If K = f∗K ′ for someK ′ ∈ D(X), then the claim results from the projection formula. In general, forfixedL, the collection ofK ∈ D(X) for which φK,L is an equivalence is closed under colimits. Asf isquasi-affine, the essential image of the pullbackf∗ : D(X) → D(U) generates the target under colimits (asthis is true for open immersions and affine maps separately),which implies the claim. �

For anyU ∈ QAff/X , we simply writeOU ∈ D(X) for the pushforward of the structure sheaf. Then theassociationU 7→ OU lets us viewQAffopp

/X as a full subcategory ofCAlg(D(X)), and one has a symmetricmonoidal identificationD(X,OU ) ≃ D(U) by Lemma2.5. Note also that ifU ⊂ X is a quasi-compactopen subset, the forgetful functorD(X,OU ) → D(X) lets us viewD(U) ≃ D(X,OU ) as the right orthog-onal ofDX\U (X). Using this, we show thatF preserves connectivity.

Lemma 2.6. F preserves connective complexes, and thusG preserves coconnective complexes.

The possiblity that Lemma2.6 could be true was suggested by an email exchange with Lurie; an earlierversion of Theorem2.1 imposed the conclusion of Lemma2.6as a hypothesis.

Proof. By adjunction, it is enough to prove the assertion forF . By approximation by perfect complexes(see Lemma2.7 below), it is enough to check thatF (K) ∈ D≤0

perf(S) if K ∈ D≤0perf(X). If not, then there

exists a points ∈ S such thatF (K)s ∈ Dperf(κ(s)) is non-connective. By replacingS with Spec(κ(s)),we may assumeS = Spec(L) for a field L. Now A := GOS ∈ CAlg(D(X)) is a field object, i.e.,D(X,A) ≃ D(Spec(L)) as a symmetric monoidal∞-category (by Lemma2.3). In particular,D(X,A)admits no non-trivial full stable subcategories closed under colimits except itself: such a category would beclosed under retracts, so it would contain the unit object, which generatesD(X,A) ≃ D(Spec(L)) undercolimits. As a special case, ifZ ⊂ X is a constructible closed subset with open complementU , then eitherDZ(X,A) = D(X,A) or A ≃ A ⊗ OU ∈ D(U). We write [A] ∈ U if the latter possibility occurs,

7

and [A] ∈ X \ U otherwise. Note that if[A] ∈ U , thenF factors throughD(X) → D(U) via a functorD(U) → D(S) that preserves compact objects (as one identifies the latterfunctor as the base change alongOU → OU ⊗ A ≃ A, and then notes that the forgetful right adjoint certainly commutes with direct sums).In the next paragraph, this will be used implicitly in arguments replacingX with U .

Choose a sequence∅ = U0 ⊂ U1 ⊂ . . . Un = X of quasi-compact opens inX such thatUi is the pushoutof an etale mapVi−1 → Ui−1 along a quasi-compact open immersionVi−1 → Spec(Ai); such presentationsalways exist (see [Sta14, Tag 08GL] or [Lur11c, Theorem 1.3.8]). LetZi = Ui \Ui−1, viewed as a reducedsubscheme (say), soZi ≃ Spec(Ai) \ Vi−1 by hypothesis. Choose the minimali such that[A] ∈ Ui. ThenF factors throughD(X) → D(Ui), so we may replaceX with Ui to assumei = n, i.e., that[A] ∈ Zn

or, equivalently, thatA ∈ DZn(X). NowDZn(X) ≃ DZn(Spec(An)) by construction (see Lemma5.12).Hence,A lifts canonically to an object ofD(Spec(An)); in fact, A ≃ A ⊗ OSpec(An). This implies thatF factors through the pullbackD(X) → D(Spec(An)). Hence, we reduce to the case whereX is affine,where everything is clear: any connective perfect complexK is then a retract of a finite colimit of finitefreeOX-modules, soF (K) has the same property onS, whenceF (K) is connective asD≤0(S) containsOS = F (OX) and is closed under retracts and colimits. �

The following lemma was used above.

Lemma 2.7. EveryK ∈ D≤0(X) may be written as a filtered colimitK = colimKi withKi ∈ D≤0perf(X).

Proof. By absolute noetherian approximation (see [CLO12, Theorem 1.2.2] or [Sta14, Tag 07SU]), we canwrite X = limXi as cofiltered limit of qcqs and finitely presentedZ-spacesXi. If fi : X → Xi is thenatural map, then the natural mapcolim f∗i fi,∗K → K is an isomorphism, so we reduce to the case whereX = Xi is noetherian. AsD(X) = Ind(Dperf(X)), anyK ∈ D(X) can be written as a filtered colimitK = colimKj with Kj ∈ Dperf(X). If K is connective, then we can also writeK = colim τ≤0Kj (asfiltered colimits are exact), soK may be expressed as a filtered colimit of connective coherentcomplexes(by the noetherian assumption). We may then assumeK is itself a bounded coherent connective complex.Fix someN > 0. We will construct a diagram

K0 → K1 → K2 → K3 → . . .

of perfect complexes inD≤0(X)/K such thatcofib(Ki → K) is (i·N)-connective. The left-completeness ofD(X) (see [Lur11d, Proposition 2.3.18]) then givescolimKi ≃ K, proving the claim. AsK is connective,we start withK0 = 0. Fix somen > 0, and assume inductively we have constructed a finite tower

K0 → K1 → K2 → · · · → Kn−1

in D≤0(X)/K such thatcofib(Ki → K) is (i ·N)-connective fori ≤ n− 1. LetQ := cofib(Kn−1 → K).Choose a connective perfect complexL and a mapL → Q with an(n ·N)-connective cofibre; this can bedone via [Sta14, Tag 08HH]. SetKn := L×Q K. This gives a map of cofibre sequences

Kn−1// Kn

//

��

L

��

Kn−1// K // Q.

Thencofib(Kn → K) is thus(n ·N)-connective. Continuing in this manner gives the desired diagram. �

Remark 2.8. As F preserves connectivity, there is an induced adjunctionQCoh(X)H0F // QCoh(S)H0Goo ,

whereH0F is the composition

QCoh(X) → D≤0(X)F→ D≤0(S)

H0

→ QCoh(S),8

whileH0G is the composition

QCoh(S) → D≥0(S)G→ D≥0(S)

H0

→ QCoh(X).

Moreover, the left adjointH0F is symmetric monoidal, while the right adjointH0G preserves filteredcolimits. It follows formally thatH0F : CAlg(QCoh(X)) → CAlg(QCoh(S)) preserves compact objects.

Recall that we are viewing bothAff/X andQAff/X as full subcategories ofCAlg(D(X)) via Lemma2.5. We check thatF preserves these subcategories, i.e., one can pullback (quasi-)affine morphisms viaF :

Lemma 2.9. F induces functorsAff/X → Aff/S andQAff/X → QAff/S . For anyU ∈ QAff/X , onehas an induced cocontinuous symmetric monoidal functorFU : D(X,OU ) → D(S,OF (U)) that preservescompact objects, is compatible withF , and carriesLU/X toLF (U)/S .

Proof. The case of affine morphisms is immediate from Lemma2.4 asAffopp/X ≃ CAlg(D≤0(X)). More-

over, in this case,F also preserves morphisms of locally almost finite presentation (see Lemma2.10). For the

quasi-affine case, fix some quasi-affine mapf : U → X. Then we can choose a factorisationUj→ U

π→ X

with π affine andj a quasi-compact open immersion. Leti : Z → U be the (constructible) closed com-plement ofU , given some finitely presented closed subscheme structure.By the affine case, we obtain analmost finitely presented closed immersionZ ′ := F (Z) → F (U) =: U ′ in Aff/S . Let U ′ := U ′ \ Z ′ bethe displayed quasi-compact open subset. Then we claim thatF carriesU to U ′, i.e., thatF (OU ) ≃ OU ′

in CAlg(D(X)). For this assertion, we may replaceX with U andS with U ′ to assume thatU ⊂ XandU ′ ⊂ S are quasi-compact open subsets with constructible closed complementsZ ⊂ X andZ ′ ⊂ Srespectively. Now note that one has a cofibre sequence

ΓZ(OX) → OX → OU

which defines another cofibre sequence

F (ΓZ(OX)) → OS → F (OU ).

We claim that this last sequence coincides with

ΓZ′(OS) → OS → OU ′ ,

which certainly implies the desired result. For this, we check that the equivalenceΦ : D(X,GOS) ≃ D(S),given by the inverse ofG, carriesDZ(X,GOS) ontoDZ′(S); one then uses the description ofΓZ → idXandΓZ′ → idS as counits of the adjunctionsDZ(X,GOS)

// D(X,GOS)oo andDZ′(S) // D(S)oo

respectively. The construction ofΦ shows thatΦ(OZ ⊗GOS) = F (OZ) = OZ′ as commutative algebras.It is thus enough to note thatDZ(X,GOS) is generated under colimits by(OZ ⊗GOS)-complexes, and thatDZ′(S) is generated under colimits byOZ′-complexes; for this, one reduces to the affine case by suitableMayer-Vietoris sequences, and then follows the proof of [Toe12, Proposition 3.10] or [Lur11a, Lemma6.17]. It remains to check thatFU (LU/X) ≃ LF (U)/S . For this, it is enough to check that the identificationD(U) ≃ D(X,OU ) carriesLU/X to LOU/OX

. If U ∈ Aff/X , then this is clear. By the transitivity cofibresequences, we reduce to showing thatLOU/OX

= 0 if U ⊂ X is a quasi-compact open. Note thatOX → OU

is an epimorphism inCAlg(D(X)): one hasOU ⊗ OU ≃ OU via base change for coherent cohomology(see [Lur11c, Corollary 1.1.3 (3)]). The base change formula for cotangent complexes [Lur11b, Proposition8.3.3.7] then showsLOU/OX

≃ 0. �

The next lemma was used earlier.

Lemma 2.10. The functorF : Aff/X → Aff/S preserves morphisms locally of (almost) finite presentation.

Proof. We first remark thatH0F : CAlg(QCoh(X)) → CAlg(QCoh(S)) preserves compact objects asH0G is compatible with filtered colimits. It follows that ifA ∈ CAlg(D≤0(X)) is locally of almostfinite presentation, thenH0F (A) is finitely presented as an ordinary algebra; here we use thatA′ ∈

9

CAlg(QCoh(X)) is a compact object if and only if the corresponding affine morphismSpec(A′) → Xis a finitely presented map of classical schemes. To handle higher homotopy groups, we use the characteri-zation of (almost) finite presentation in terms of cotangentcomplexes in the presence on finite presentationat the classical level (see [Lur11b, Theorem 8.4.3.18]). �

Recall that a mapg : U → V of qcqs algebraic spaces is etale if and only ifLU/V ≃ 0 andg is locally ofalmost finite presentation.

Lemma 2.11. The functorF : QAff/X → QAff/S preservesetale morphisms.

Proof. This is immediate from Lemma2.9. �

We also have:

Lemma 2.12. The functorF : QAff/X → QAff/S preserves finite limits andetale surjections.

Proof. The preservation of finite limits follows from the symmetricmonoidal assumption onF , togetherwith the fact that the fully faithful functorQAffopp

/X → CAlg(D(X)) given byU 7→ OU preserves finitecolimits (which comes from base change for coherent cohomology). Now assumef : U → V is an etalemap. Thenf is surjective if and only ifD(X,OV ) → D(X,OU ) is conservative. Thus, it is enough to checkthat for surjectivef , the induced functorD(S,OF (V )) → D(S,OF (U)) is conservative. For this, considerthe commutative diagram

D(X,OV ) //

��

D(X,GOS ⊗ OV )≃ //

��

D(S,OF (V ))

��

D(X,OU ) // D(X,GOS ⊗ OU )≃ // D(S,OF (U)).

The second vertical arrow is simplyK 7→ K ⊗OVOU , which is conservative by hypothesis. Hence, the last

vertical arrow is also conservative, as wanted. �

We can now put the above ingredients together.

Proof of Theorem.Note first that the theorem is true forX affine (by [Lur11d, Theorem 3.4.2] and Lazard’stheorem that flat modules are ind-(finite free), for example). In general, we may choose an etale hypercoverπ∗ : U∗ → X with eachU i affine, soU i → X is quasi-affine. ThenF (U∗) → S is an etale hypercover byquasi-affineS-schemes by Lemma2.12. By the affine case, there is a mapf : F (U∗) → U∗ of simplicialschemes such that the pullbackf∗ : D(X,OU i) → D(S,OF (U i)) coincides withFU i . Under the etaledescent identificationsD(X) = TotD(X,OU∗) andD(S) ≃ TotD(S,OF (U∗)), one hasTotFU i ≃ F . Itfollows that|f | : |F (U∗)| → |U∗| is the desired mapS → X. �

Remark 2.13. The previous results give us an identificationHom(S,X) ≃ FunL⊗(D(X),D(S)), and afully faithful embeddingHom(S,X) ⊂ FunL⊗(QCoh(X),QCoh(S)) for qcqs algebraic spaces. We do notknow if the latter is an equivalence: it is not clear if everyF ∈ FunL⊗(QCoh(X),QCoh(S)) preservesthe subcategory of finitely presented quasi-coherent sheaves (= the subcategory of compact objects)4. Theidentification of compact objects with dualizable objects inD(X) solves this problem in the derived setting.

4Gabber has informed us that this obstruction is the only one.10

3. THE CASE OF SCHEMES, REVISITED

Brandenburg and Chirvasitu [BC12] have shown the following:

Theorem 3.1. For qcqs schemesS andX, one hasHom(S,X) ≃ FunL⊗(QCoh(X),QCoh(S)).

For convenience, we recall the key points of their proof below.

Proof. We first prove full faithfulness ofHom(S,X) → FunL⊗(QCoh(X),QCoh(S)). Sayf, g ∈ Hom(S,X)admit a symmetric monoidal natural transformationη : f∗ → g∗; it follows thatη lifts to a natural trans-formation of the two induced functorsCAlg(QCoh(X)) → CAlg(QCoh(S)). We will showf = g andη = id. Assume first thatS andX are affine. Then the mapηOX

: OS → OS is a ring homomor-phism inQCoh(S), and hence the identity. AsQCoh(X) is generated byOX under colimits, the claimfollows in this case. In general, the claim is local onS. Moreover, for any closed subsetZ ⊂ X, the mapηZ : Of−1(Z) → Og−1(Z) is aOS-algebra map, sog−1(Z) ⊂ f−1(Z). In particular, we may coverS byaffine opensSi such that bothf |Si

andg|Sifactor through some affine openUi ⊂ X. By replacingS with

each element of such a cover, we may assume bothf andg factor through an affine openj : U → X. Bothf∗ andg∗ then factor throughj∗ as cocontinuous symmetric monoidal functors; here one usesj∗j∗ ≃ id.Moreover, one checks thatη induces a symmetric monoidal natural transformation of theresulting twofunctors. Thus, by replacingX with U , we reduce to the affine case treated earlier.

For essential surjectivity, fix some functorF . As Hom(−,X) andFunL⊗(QCoh(X),QCoh(−)) arefpqc stacks, we may assumeS is affine. IfX is affine, the claim is clear. In general, for every closedsubschemeZ ⊂ X, one has a closed subschemef−1(Z) ⊂ S defined viaF (OZ) = Of−1(Z) with functorsQCoh(Z) → QCoh(f−1(Z)) andQCohZ(X) → QCohf−1(Z)(S). If Z is constructible with an affinecomplementU ⊂ X, andV ⊂ S \ Z ′ is some affine open, one has an induced cocontinuous symmetricmonoidal functorQCoh(U) → QCoh(V ). As U andV are affine, this arises as pullback along a mapfV,U : V → X factoring throughU . Using full faithfulness, it is easy to check that the collection {fV,U}of maps thus obtained are compatible. It is thus enough to check the collection of allV ’s obtained by thisprocedure coverS. If not, there exists somes ∈ S such thats ∈ f−1(Z) for all Z ⊂ X closed. Choose anaffine open cover{U1, . . . , Un} ofX with complementsZi := X \Ui. Then⊗n

i=1OZi= 0 as∪iUi = X, so

⊗ni=1Of−1(Zi) = 0 as well. On the other hand,Of−1(Zi)⊗κ(s) 6= 0, so the tensor product⊗n

i=1Of−1(Zi) 6= 0as well (since tensor products of non-zero vector spaces arenon-zero), which is a contradiction. �

Recall that a qcqs schemeX is said to haveenough vector bundlesif every finitely presented quasi-coherent sheaf can be expressed as the cokernel of a map of vector bundles; any scheme that is quasi-projective over an affine is an example, and the class of such schemes is closed under cofiltered limits withaffine transitions. For such schemes, one may go even furtherthan Theorem1.7

Corollary 3.2. LetX andS be qcqs schemes. AssumeX has enough vector bundles. Then

Hom(S,X) ≃ FunL⊗(Vect(X),Vect(S)).

HereFunL⊗(Vect(X),Vect(S)) refers to category of all symmetric monoidal functorsVect(X) → Vect(S)that are right exact; by duality, such functors preserve allexact sequences of vector bundles. The proof belowentails building certain functors out ofQCoh(X) starting with functors out ofVect(X); a more systematicapproach is discussed in§3.1.

Proof. We knowHom(S,X) = FunL⊗(QCoh(X),QCoh(S)), so we will identify the right hand side withFunL⊗(Vect(X),Vect(S)). Any symmetric monoidal functorF : QCoh(X) → QCoh(S) preserves vectorbundles (as these are the dualizable objects; see Lemma3.3), and thus induces a symmetric monoidal functorφ(F ) : Vect(X) → Vect(S) that preserves surjections. This construction gives a functor

φ : FunL⊗(QCoh(X),QCoh(S)) → FunL⊗(Vect(X),Vect(S)).11

Next, we claim that anyF ∈ FunL⊗(QCoh(X),QCoh(S)) is a left Kan extension of its restriction

ψ(F ) : Vect(X)φ(F )→ Vect(S)

i→ QCoh(S).

This will prove thatφ is fully faithful. To see this, it is enough to note thatVect(X) ⊂ QCoh(X) is a fullsubcategory that generatesQCoh(X) under colimits (as every finitely presented quasi-coherentsheaf is acokernel of a map of vector bundles, by assumption).

It remains to check thatφ is essentially surjective. GivenG ∈ FunL⊗(Vect(X),Vect(S)), we will builda cocontinuous symmetric monoidal functorF : QCoh(X) → QCoh(S) extendingG. For this, we firstextend toQCohfp(X), so fix someQ ∈ QCohfp(X). Given a “resolution”E• of Q, i.e., an exact sequence

E2 → E1 → Q→ 1

with Ei ∈ Vect(X), we setF (Q) := coker(F (E2) → F (E1)) ∈ QCohfp(S). We will show that thisconstruction is well-defined (i.e., independent ofE• up to unique isomorphism) and functorial inQ. Notefirst that if Q ∈ Vect(X), thenF (Q) = G(Q) by the assumption onG. To show well-definedness ingeneral, fix a second resolutionG• ofQ and a surjective mapφ• : E• → G• of resolutions; here “surjective”simply means that the mapφi : Gi → Ei is surjective for eachi. Then a diagram chase and the assumptiononG show thatφ• induces an isomorphism

φ∗ : coker(F (E2) → F (E1)) ≃ coker(F (G2) → F (G1)).

Note thatφ∗ is defined using onlyφ1, but the existence of aφ2 is needed to get a well-defined map. As anytwo resolutions can be dominated (in the sense of surjections) by a common third one, it follows thatF (Q)is well-defined up to isomorphism.

We next show thatF (Q) is well-defined up to unique isomorphism, i.e., the isomorphism φ∗ above isindependent of mapφ• chosen. Indeed, assume we have two mapsφ•, ψ• : E• → G• of resolutions. Toshow that the induced maps

φ∗, ψ∗ : coker(F (E2) → F (E1)) → coker(F (G2) → F (G1))

are the same, we can always replace the resolutionE• by one mapping surjectively onto it (by the argumentused to showF (Q) was well-defined up to isomorphism). After doing such a replacement, we can assumethat the two mapsφ1, ψ1 : E1 → G1 differ by a map lifting toG2. In this case, the two induced maps

F (φ1), F (ψ1) : F (E1) → F (G1)

differ by a map lifting toF (G2) by functoriality ofF in Vect(X), and thus the resulting two maps

φ∗, ψ∗ : coker(F (E2) → F (E1)) → coker(F (G2) → F (G1))

are visibly the same, which proves thatF (Q) is well-defined up to unique isomorphism.Next, we make this construction is functorial inQ. Given a maph : Q1 → Q2, one finds a resolutionE•

of Q1,G• of Q2, and a mapφ• : E• → G• lifting h. This defines a mapφ∗ : F (Q1) → F (Q2). Using thetrick used to show well-definedness ofF (Q) above, one checks thatφ∗ is independent ofE•, G•, andφ•.Thus, the constructionQ 7→ F (Q) is functorial inQ, so we obtain a functorF : QCohfp(X) → QCoh(S)which extendsG, and carries resolutions as above to right exact sequences.As one can lift right exactsequences inQCohfp(X) to right exact sequences of resolutions, it follows thatF is right exact, so wehave produced a finitely cocontinuous functorF : QCohfp(X) → QCoh(S) extendingG. By passing toinductive limits, one obtains a cocontinuous functorF : QCoh(X) → QCoh(S) extendingG. We leave itto the reader to check that one may endowF with the structure of a symmetric monoidal functor extendingthe given one onG in a unique (and evident) way, which is enough to prove the desired claim. �

Lemma 3.3. LetX be a qcqs scheme. ThenE ∈ QCoh(X) is dualizable if and only ifE is a vector bundle.12

Proof. It is clear that vector bundles are dualizable. Conversely,assumeE ∈ QCoh(X) is dualizable withdualE∨. To showE is a vector bundle, by localising, we may assumeX = Spec(A) is affine. We nowidentifyQCoh(X) with ModA to solve the corresponding question for modules. ThenHom(E,−) = E∨⊗(−) commutes with filtered colimits, and thusE is finitely presented. Similarly,Hom(E∨,−) = E ⊗ (−),soE is flat. Any finitely presented flatA-module is finite locally free, proving the claim. �

One may wonder if the cocontinuity condition on the functorsappearing on the right hand side of Corol-lary 3.2 is automatically satisfied: the next two examples show this is not the case, and that such functorsabound in nature. Moreover, these examples also indicate a potential subtlety in the applying Corollary3.2:the condition that a map inVect(S) be surjective is defined in terms of the ambient categoryQCoh(S),and is not intrinsic to the categoryVect(S). One may raise similar objections to Theorem2.1, but theyare easily refuted: it is almost impossible (certainly quite unnatural) to write down a non-exact functorDperf(X) → Dperf(S), and the exactness condition is intrinsic to the∞-categories of perfect complexes.

Example 3.4.LetX be an affine regular noetherian scheme of dimension2, and letx ∈ X be a closed point.SetS = X \ {x}. Then the inclusionj : S → X induces an equivalencej∗ : Vect(X) → Vect(S) by theAuslander-Buschbaum formula; explicitly, for anyE ∈ Vect(S), the double dualE

∗∗of anyE ∈ Coh(X)

extendingE is a vector bundle extendingE. However, the mapj is certainly not an isomorphism. This doesnot contradict Corollary3.2as the symmetric monoidal equivalenceVect(S) ≃ Vect(X) doesnotpreservesurjections: the inverse toj∗ is given by reflexivising a coherent extension, and the reflexivisation processloses surjectivity properties at a missing point. Explicitly, if X = A2 = Spec(k[y, z]) over a fieldk and

x = (0, 0), then the mapO⊕2X

(y,z)→ OX in Vect(X) is surjective overS, but not atx.

Example3.4 might lead one to suspect that such phenomenon can be avoidedin the projective case.However, this is not the case:

Example 3.5. Fix a fieldk, and letX = P1 andS = Spec(k). We will construct a symmetric monoidalfunctor F : Vect(P1) → Vect(S) which does not come fromk-point of P1. Our functorF will notpreserve surjections. To constructF , consider the natural mapπ : A2 \ {0} → P1. Thenπ∗ : Vect(P1) →Vect(A2−{0}) is certainly symmetric monoidal. As in Example3.4, we knowj∗ : Vect(A2) → Vect(A2\{0}) is an equivalence, wherej : A2 \ {0} → A2 is the defining map. Thus, we find a symmetric monoidalfunctorF : Vect(P1) → Vect(k) given viaF = i∗ ◦ (j∗)−1 ◦ π∗, wherei : S → A2 is the origin. Onecan check easily thatF does not come from geometry. In fact, the surjectionO

⊕2X → OX(1) (definingP1)

is carried byF to the0 mapO⊕2S → F (OX(1)) ≃ OS .

3.1. RecoveringQCoh from Vect. Fix a qcqs schemeX with enough vector bundles. Examples3.4 and3.5 show that there is no way to recoverQCoh(X) from Vect(X) as there is no hope knowing what “sur-jective” maps should be intrinsically in terms ofVect(X). We sketch now why this is the only obstruction:one functorially recoversQCoh(X) from Vect(X) equipped with (the extra data of) the class of “surjec-tive maps.” The ideas below already appear in the proof of Corollary 3.2 implicitly, and will not be usedelsewhere in the paper. We begin by defining the ambient category where all constructions will take place.

Definition 3.6. Let Vect(X) be the category of additive presheavesVect(X)opp → Ab onVect(X).

The mapVect(X) → Vect(X) enjoys a good universal property.

Lemma 3.7. The categoryVect(X) is a cocomplete abelian category, and the Yoneda embeddingVect(X) →

Vect(X) is the universal additive mapVect(X) → A to a cocomplete abelian category.

Proof. Left to the reader (see [KS05, Proposition 3.6] as well as [DL07]). �

Roughly speaking, we view the extra data of the class of surjective maps inVect(X) as a topology onVect(X); the categoryVect(X) is then the category of presheaves, while the category of interest will be the

13

category for sheaves. However, to avoid discussing topologies on additive categories, we encode the data ofsurjections in terms of their coequalizers to isolate the objects of interest.

Definition 3.8. Let S be the class of maps inVect(X) of the formcoker(E ×Q E

p1−p2→ E

)→ Q, where

E → Q is a surjective map inVect(X). Let C ⊂ Vect(X) be the full subcategory ofS-local presheaves,

i.e., presheavesF such that1 → F (Q) → F (E)p1−p2→ F (E ×Q E) is exact for every surjectionE → Q

or, equivalently, thatF (Q) ≃ F (T ) for any mapT → Q in S.

The Yoneda embeddingVect(X) → Vect(X) lands inC, and the basic properties ofC, summarized inthe next lemma, are close analogues of the usual properties of sheaves on a coherent site.

Lemma 3.9. C is closed under limits and filtered colimits inVect(X). The inclusioni : C → Vect(X)

admits an exact left adjointL : Vect(X) → C. In particular,C admits all limits and colimits.

Proof. We first define a functorL+ : Vect(X) → Vect(X) via the familiar formula from sheaf theory, i.e.,

L+(F )(Q) = colim ker(F (E) → F (E ×Q E)),

where the colimit is indexed by the (opposite of the) category I(Q) of surjective mapsE → Q. Using theclassS introduced above, we can rewrite this as

L+F (Q) = colimF (T ),

where the colimit is indexed by the (opposite of the) category J(Q) of mapsT → Q in S. Note that eachmap inS is a monomorphism, soJ(Q) is a poset, and a subposet of the posetSub(Q) of subobjects ofQ in Vect(X). As a fibre product of bundle surjections defines a square of bundle surjections,J(Q) isstable under finite intersections inSub(Q), so the second colimit definingL+(F )(Q) is filtered. Let ustemporarily callF ∈ Vect(X) separatedif F (Q) → F (E) is injective for anyE → Q in I(Q). Using thesecond description, one checks thatL+(F ) is always separated for anyF ∈ Vect(X). Moreover, ifF isseparated, then a diagram chase shows thatL+(F ) ∈ C. In particular, the functorL+ ◦L+ can be viewed asa functorL : Vect(X) → C. It is also clear thatL(F ) = L+(F ) = F if F ∈ C, and one then checks thatLgives a left adjoint toi (using that any mapF → G withG ∈ C extends canonically to a mapL+(F ) → G).Finally, the second description ofL+ given above shows thatL+ is exact, and hence so isL. �

As in sheaf theory, the left-adjointL : Vect(X) → C is a localization ofVect(X).

Lemma 3.10. Any cocontinuous functorG : Vect(X) → A to a cocomplete abelian categoryA whichcarries maps inS to isomorphisms factors uniquely overL, i.e.,T ≃ T ◦ i ◦ L.

Proof. For notational convenience, letM = i ◦ L. It is enough to show that for anyF ∈ Vect(X), themapF → M(F ) is in the smallest strongly saturated classS of maps inVect(X) containingS; herea subcategoryS ⊂ Mor([1], Vect(X)) is called a strongly saturated class if it is closed under colimits,satisfies the2-out-of-3 property, and if maps inS are stable under arbitrary pushouts inVect(X); see[Lur09, Definition 5.5.4.5] for more. Indeed, for anyG : Vect(X) → A as in the lemma, the collection ofmapsf in Vect(X) such thatG(f) is an isomorphism is a strongly saturated class containingS, and thusGcarriesS to isomorphisms, soG(F ) ≃ G(M(F )) as wanted. To finish, it is enough to noteF → M(F ) isin S by general facts about Bousfield localizations; see [Lur09, Proposition 5.5.4.15]. �

This leads to a simple universal property describing the Yoneda embeddingVect(X) → C.

Corollary 3.11. Any exact additive functorF : Vect(X) → A to a cocomplete additive categoryA extendsuniquely to a cocontinuous functorC → A.

14

Proof. By Lemma3.7, any suchF extends uniquely to a cocontinuous functorF : Vect(X) → A. Weclaim thatF carries maps inS to isomorphisms. To see this, fix a surjectionE → Q of bundles. The kernelK is a bundle as well, so we obtain a short exact sequence

1 → K → E → Q→ 1

in Vect(X). AsF is exact, the sequence

1 → F (K) → F (E) → F (Q) → 1

is also exact. Using the identificationE×QE ≃ E×K, it follows thatF (E×QE) ≃ F (E)×F (Q) F (E).

This implies thatF carries the mapcoker(E ×Q Ep1−p2→ E) → Q in S to an isomorphism. Lemma3.10

then shows thatF factors through a cocontinuous functorC → A, proving the claim. �

The categoryC constructed above is actually a familiar object.

Proposition 3.12. The functorQCoh(X) → Vect(X) given byF 7→ Hom(−, F ) factors through anequivalenceQCoh(X) ≃ C.

Proof. The associationF 7→ Hom(−, F ) gives a continuous functorQCoh(X) → Vect(X) that factorsthroughC, thus giving a continuous functorΦ : QCoh(X) → C. We first check thatΦ is cocontinuous. Forthis, it is enough to check thatΦ preserves filtered colimits and finite colimits separately.For finite colimits,we must checkΦ is right exact. IfK → F → Q → 1 is a right exact sequence inQCoh(X), then the onlynon-trivial claim is thatΦ(F ) → Φ(Q) is surjective inC. By definition ofC, we must check that for anymapE1 → Q with E1 ∈ Vect(X), there exists a surjectionE2 → E1 such that the compositeE2 → Qlifts to F ; this follows from the existence of enough vector bundles onX, the fact that vector bundles arecompact inQCoh(X), and approximation by finitely presented quasi-coherent sheaves (see Lemma6.11).To show preservation under filtered colimits, asi : C → Vect(X) preserves filtered colimits, it is enough tocheck that the compositeQCoh(X) → Vect(X) preserves filtered colimits; this is a consequence of objectsin Vect(X) being compact in bothQCoh(X) andVect(X).

To showΦ is an equivalence, we construct an inverseΨ : C → QCoh(X). The canonical inclusionVect(X) → QCoh(X) is exact. By the universal property in Lemma3.11, this inclusion factors through acocontinuous functorΨ : C → QCoh(X). As bothΦ andΨ are cocontinuous, and because bothQCoh(X)andC are generated under colimits byVect(X), it is enough to check thatΨ ◦ Φ andΦ ◦Ψ are the identityonVect(X), which is obvious. �

This gives a characterization ofQCoh(X) in terms ofVect(X) equipped with the notion of surjections.

Corollary 3.13. Any exact additive functorVect(X) → A to a cocomplete abelian categoryA factorsuniquely through a cocontinuous functorQCoh(X) → A.

15

4. ALGEBRAIZATION OF JETS

The goal of this section is to prove the following theorem.

Theorem 4.1. Let A be a ring equipped with an idealI such thatA is I-adically complete. Fix a qcqsalgebraic spaceX. ThenX(A) ≃ limX(A/In) via the natural map.

Proof. We first prove injectivity, though this step could be avoidedby using the argument in the next para-graph together with Proposition5.1(which is independent of this§). Fix two mapsf, g : Spec(A) → X thatinduce the same mapǫn : Spec(A/In) → X. LetZ → Spec(A) be the pullback of∆ : X → X×X along(f, g) : Spec(A) → X×X. ThenZ → Spec(A) is a quasi-compact monomorphism with a specified systemof compatible sections over eachSpec(A/In). In particular,Z is quasi-affine. By Example4.8 (or simplythe next paragraph), one obtains a mapSpec(A) → Z such that the compositeSpec(A) → Z → Spec(A)agrees with the identity moduloIn for all n. In other words,Z → Spec(A) is a quasi-compact monomor-phism with a section, and hence an isomorphism, which provesf = g.

For surjectivity, fix a compatible system of maps{ǫn : Spec(A/In) → X}. Pullback gives a compat-ible system{ǫ∗n : Dperf(X) → Dperf(A/I

n)} of exact symmetric monoidal functors, and thus an exactsymmetric monoidal functor

F : Dperf(X) → limDperf(A/In).

By Lemma4.2 for the special caseIn := In, the canonical mapDperf(A) → limDperf(A/In) is an

equivalence, soF may be viewed as an exact symmetric monoidal functorDperf(X) → Dperf(A). ByTheorem2.1, this comes from a unique mapǫ : Spec(A) → X with F ≃ ǫ∗. It is clear by construction thatǫ extends eachǫn (asF extendsǫ∗n), which gives the surjectivity ofX(A) → limX(A/In). �

The following patching result for perfect complexes is a crucial ingredient in the proof above.

Lemma 4.2. LetA be a ring equipped with a descending sequence{. . . In ⊂ · · · ⊂ I2 ⊂ I1} of ideals.Assume thatA ≃ limA/In, and thatIn/In+1 is nilpotent for eachn. ThenDperf(A) ≃ limDperf(A/In).

Proof. The full faithfulness ofDperf(A) → limDperf(A/In) is easy to check using thatK ⊗A − com-mutes with limits ifK ∈ Dperf(A). The essential surjectivity is standard; see [Sta14, Tag 09AW] or [Fal99,Corollary 12]. The key point is represent an object{Kn} ∈ limDperf(A/In) by a projective system{Pn}of complexes of finite projectiveA/In-modules such that the transition mapPn+1 → Pn induces an iso-morphismPn+1/In ≃ Pn of chain complexes (not merely in the derived category); this can be done byinduction onn using the nilpotence assumption and [Sta14, Tag 09AR]. �

Remark 4.3. As pointed out by Nicaise, Theorem4.1and its proof apply to anyadmissibletopological ringA, i.e., a ringA equipped with ideals{In} as in Lemma4.2. An elementary example of such a ring, whichis not adic as in Theorem4.1, is: A = kJx, yK with Ik = (x · yk). An example that is more relevant inp-

adic geometry comes from crystalline cohomology:A := Fp〈x〉, the completed divided power polynomial

algebra on a generatorx, with Ik = 〈x〉[k] = (γn(x))n≥k being thekth-level of the Hodge filtration.

Theorem4.1immediately implies the following result identifying jet spaces and the Greenberg functor atinfinite level as honest mapping functors.

Corollary 4.4. LetX be a qcqs space over some base ringA.

(1) The arc space functorR 7→ limX(R[t]/(tn)) is isomorphic to the functorR 7→ X(RJtK) on thecategory ofA-algebras.

(2) If A = k is a perfect fieldk of characteristicp > 0, the infinite level Greenberg functorR 7→limX(Wn(R)) is isomorphic to the functorR 7→ X(W (R)) on the category ofk-algebras.

Proof. (1) is immediate from Theorem4.1, while (2) follows from Remark4.3as the kernel ofWn+1(R) →Wn(R) is nilpotent. �

16

Remark 4.5. Theorem4.1 implies (and is equivalent to) an existence result for sections. LetA be anI-adically complete ring for some idealI, setS = Spec(A), and fix a qcqs mapν : Y → S of algebraicspaces. If we writeX for the formalI-adic completion of anS-spaceX, then Theorem4.1is equivalent to

HomS(S, Y ) ≃ HomS(S, Y ).

In particular, every formal section ofν is automatically algebraic. Even whenA is noetherian, such resultstypically impose stronger hypotheses on the diagonal: [Gro61, Theorem 5.4.1] requiresπ to be separated,while [Lur11c, Theorem 5.4.1] requiresπ to have affine diagonal. By Example5.5, a similar remark alsoapplies toHomS(X,Y ), whereX is a properA-space withA noetherian. Moreover, using the trick inRemark4.6, if one works exclusively with schemes, one can drop all conditions on the diagonal: one has

HomS(X,Y ) ≃ HomS(X, Y )

for a properS-schemeX and an arbitraryS-schemeY , i.e., [Gro61, Theorem 5.4.1] is true for allY .

Remark 4.6(Gabber). Theorem4.1applies to any schemeX. Indeed, to proveX(A) = limX(A/In), wemay assumeX is quasi-compact. Any quasi-compact scheme is areasonablealgebraic space in the senseof [Sta14, Tag 03I8], i.e., there exists a surjective etale mapU → X with bounded fibre degree5, whereUis an affine scheme. By inspection of the proof of [Sta14, Tag 03K0], we can writeX = colimXi wherethe colimit is filtered and computed in the category of fpqc sheaves,Xi → X is a local isomorphism (inparticular, it is etale), andXi is qcqs. Given a compatible system{ǫn : Spec(A/In) → X}, there existssomei such thatǫ1 lifts to a mapǫ1 : Spec(A/I) → Xi. By the infinitesimal lifting property, each mapǫn factors through a unique mapǫn : Spec(A/In) → Xi lifting ǫ1. By the qcqs case, one finds a mapǫ : Spec(A) → Xi extending eachǫn. Composing back down toX then shows thatX(A) → limX(A/In)surjective. For injectivity, one repeats the trick involving the diagonal in the proof of Theorem4.1, using thesurjectivity just proven in lieu of Example4.8. More generally, the same argument applies to any Zariskilocally reasonable algebraic space. Similarly, Corollary4.4also extends to all schemes.

Remark 4.7. Theorem4.1and Remark4.6show: ifA is a ring which isI-adically complete for some idealI, thencolim Spec(A/In) ≃ Spec(A) in the category of schemes. This gives an instance of the generalphenomenon that colimits in schemes can be quite different from the corresponding colimit in fpqc sheaves.Indeed, the colimit of the diagram{Spec(A/In)} as an fpqc sheaf overSpec(A), which may be viewedas an ind-scheme, has noA-points unlessI is nilpotent. In particular, this shows that (the perfect complexcomponent of) Theorem2.1does not extend to ind-schemes.

We give a few examples where Theorem4.1can be proved directly by classical methods.

Example 4.8.AssumeX is quasi-affine. Then the affine case immediately shows thatX(A) → limX(A/In)is injective. For surjectivity, fix mapsǫn : Spec(A/In) → X compatible inX. Choose an affineY con-taining X as a quasi-compact open with constructible closed complement Z. Then the compsite mapsµn : Spec(A/In) → Y extends to a unique mapµ : Spec(A) → Y . ForK ∈ DZ,perf(Y ), one has

µ∗K = limµ∗nK = 0,

where the first equality uses thatµ∗K is perfect, so−⊗OSpec(A)µ∗K commutes with limits, while the second

equality is a simple consequence ofµn factoring throughX. Writing OZ as a colimit of suchK then impliesµ∗OZ = 0, i.e.,µ−1(Z) = ∅, which implies thatµ factors through a mapǫ : Spec(A) → X, as desired.

Example 4.9. AssumeX = Pm. A compatible system{ǫn : Spec(A/In) → X} of maps is determinedby a compatible system of objects{Ln, sn,0, . . . , sn,m} comprising an invertible(A/In)-moduleLn and

elementssn,i ∈ Ln such that the induced map(A/In)⊕m sn,i→ Ln is surjective. ThenL := limLn is an

invertibleA-module by Lemma4.11, andsi := lim sn,i is an element ofL such thatA⊕m si→ L is surjective

5A typical unreasonable example isA1C/Z, whereZ acts by translation.

17

(by Nakayama). This gives a mapǫ : Spec(A) → X lifting ǫn for eachn. One checks thatǫ is the uniquesuch extension, which provesX(A) ≃ limX(A/In) in this case. More generally, the same argumentapplies to any scheme that is quasi-projective over an affine.

Example 4.10. AssumeA is noetherian, andX is separated. Then we can writeX = limXi with Xi

finitely presented separatedZ-schemes, and all transition mapsXi → Xj affine. As the assertionX(A) =limX(A/In) is compatible with inverse limits inX, we may reduce to the caseX = Xi is a finitely

presentedZ-scheme. NowlimX(A/In) is exactly the set of sections of the map X × Spec(A) → Spf(A)obtained as the formalI-adic completion of the mapX × Spec(A) → Spec(A). By formal GAGA, whichapplies as bothA andX are noetherian, each such section is (uniquely) algebraizable by [Sta14, Tag 0899],which immediately givesX(A) ≃ limX(A/In).

The following lemma was used Example4.9and mentioned in the introduction.

Lemma 4.11.LetA be a ring,I ⊂ A an ideal, and assumeA ≃ limA/In. ThenVect(A) ≃ limVect(A/In).

Proof. This follows from Lemma4.2using [Lur11d, Corollary 2.7.33] to identifyVect(A) as the dualizableobjects ofDperf(A). Alternatively, we can argue directly as follows. The natural functorF : Vect(A) →limVect(A/In) is fully faithful on finite free modules by assumption. By passage to summands, it followsthatF is fully faithful. For essential surjectivity, we must check: if {Pn} ∈ limVect(A/In), thenP :=limPn is a finite projectiveA-module. Fix a projectorǫ1 ∈MN (A/I) cutting outP1, i.e., we fix a surjectionf1 : (A/I)

⊕N → P1 as well as a sections1 : P1 → (A/I)⊕N such thats1◦f1 = ǫ1. Lifting sections gives acompatible system{fn : (A/In)⊕N → Pn} of surjections. By using the projectivity ofPn overA/In, oneinductively constructs a compatible system of sections{sn : Pn → (A/In)⊕N} of {fn}. The composition{ǫn := sn ◦ fn} is a projector inMN (A) = limMN (A/In) that definesP , soP is projective. �

We also give an example showing that Theorem4.1does not extend to all algebraic stacks.

Example 4.12. LetA/C be an abelian variety, and consider the classifying stackX = B(A). ThenX(R)is the groupoid ofA-torsors overR, for any ringR. We will construct a geometrically regularC-algebraR which is complete along an idealI, and a compatible system ofA-torsorsYn → Spec(R/In) suchthat the order of eachYn in H1(Spec(R/In), A) is infinite. Given such data, if there exists anA-torsorY → Spec(R) lifting eachYn, then the order ofY would have to be∞, which cannot happen: the groupH1(Spec(R), A) is torsion asR is regular by [Ray70, Proposition XIII.2.6]. It follows that the family{Yn}defines a point oflimX(R/In) that does not lift toX(R). To construct this data, takeR to be the completionof A2 along an irreducible nodal cubicC ⊂ A2 with Spec(R/I) = C. Note thatSpec(R) is geometricallyregular overA2, and hence certainly so overSpec(C). The enlarged fundamental pro-group ofSpec(R/I)admitsZ as a quotient. Choosing a non-torsion pointP ∈ A(C) then gives anA-torsorY1 → Spec(R/I)with infinite order; explicitly, we glue the trivialA-torsor over the normalisation ofC to itself using trans-lation byP as the isomorphism over the node. The mapH1(Spec(R/In+1), A) → H1(Spec(R/In), A) issurjective (in fact, bijective) as the obstruction to lifting anA-torsorYn → Spec(R/In), viewed as a mapfn : Spec(R/In) → B(A), to anA-torsorYn+1 → Spec(R/In+1) is an element of

Hom(f∗nLB(A), In/In+1[1]) ≃ H2(Spec(R/I), ω∨

Y1⊗ In/In+1),

which is0. Thus, we find the desired family{Yn} by inductively lifting.

The construction in Example4.12can be modified to show that Theorem4.1 fails for Artin 2-stacks.

Example 4.13. SetX = K(Gm, 2). Thenπ0(X(S)) = H2(Spec(S),Gm) for any ringS. We will con-struct a geometrically regularC-algebraR that is complete along an idealI such thatH2(Spec(R/In),Gm) ≃H2(Spec(R/I),Gm) contains a point of infinite order. AsR is regular, one knowsH2(Spec(R),Gm) istorsion by a standard result in etale cohomology (see [Mil80, Corollary IV.2.6]), so the mapX(R) →limX(R/In) is not essentially surjective. For the construction, takeR0 to be the glueing two copies of

18

A2 along a curveC ⊂ A2 of geometric genus≥ 1 overC. ThenPic(C) ⊂ H2(Spec(R0),Gm) via asuitable Mayer-Vietoris sequence, soH2(Spec(R0),Gm) certainly contains points of infinite order. SetR to be completion of a suitable surjectionP → R0 with P a polynomial ring, andI = ker(R → R0).Then this pair(R, I) satisfies the desired properties:R is geometrically regular by construction, and onehasH2(Spec(R/In),Gm) = H2(Spec(R/I),Gm) = H2(Spec(R0),Gm) by deformation theory.

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5. FORMAL GLUEING

The goal of this section is to revisit the Beauville-Laszlo theorem [BL95] and variants, and prove Theorem1.4. We begin by noting that the proof of Theorem4.1 actually gives a criterion for deciding when a qcqsalgebraic space is a colimit of a diagram of such spaces.

Proposition 5.1. LetX be a qcqs algebraic space, and let{Xi} be a diagram of qcqsX-spaces. Assumethat D(X) ≃ limD(Xi) via the natural maps. ThenDperf(X) ≃ limDperf(Xi), and this implies thatX = colimXi in the category of qcqs algebraic spaces.

Proof. For the first part, we use thatDperf(S) is exactly the category of dualizable objects inD(S) for anyqcqs algebraic spaceS by [Lur11a, Lemma 8.16]. The claim follows as duals are computed “pointwise” inlimit of symmetric monoidal∞-categories by [Lur11b, Proposition 4.2.5.11]. The second part now followsfrom Theorem2.1as the natural mapDperf(X) → limDperf(Xi) is exact and symmetric monoidal. �

Remark 5.2. Let X and {Xi} be as in Proposition5.1. If D(X) ≃ limD(Xi), thenX ≃ colimXi

as above. However, one can say a bit more: these colimits areuniversalwith respect to flat qcqs mapsg : Y → X, i.e., one hasY ≃ colim Y ×X Xi via the natural map. To prove this, it suffices to show thatD(Y ) ≃ limD(Y ×X Xi) via the natural map. By Lurie’s version [Lur11a, Corollary 8.22] of [BZFN10]and flatness, we knowD(Y ×X S) ≃ D(Y )⊗D(X)D(S) for any qcqsX-spaceS; we note here that a qcqsalgebraic space is perfect in the sense of [Lur11a, Definition 8.14] by [Lur11c, Corollary 1.5.12]. Using thisbase change formula, a formal argument involving the diagonal ∆ : Y → Y ×X Y shows thatD(Y ) is self-dual as aD(X)-module: the mapcoev : D(X) → D(Y ×X Y ) is∆∗◦g

∗, whileev : D(Y ×X Y ) → D(X)is g∗ ◦∆∗ (see [BZFN10, Corollary 4.7]). In particular, the functor−⊗D(X) D(Y ) is continuous, so

D(Y ) ≃ D(Y )⊗D(X) limD(Xi) ≃ lim(D(Y )⊗D(X) D(Xi)

)≃ limD(Y ×X Xi),

which proves the stability of these colimits under flat base change.

Remark 5.3. The implication(Dperf(X) ≃ limDperf(Xi)

)⇒

(X ≃ colimXi

)in Proposition5.1

cannot be reversed. For example, consider a non-trivial finite groupG and letX• be the usual simplicialscheme presentingBG over a fieldk. Then the colimit ofX• as an algebraic space is simplySpec(k). Onthe other hand,limDperf(X•) ≃ Dperf(BG) is notDperf(Spec(k)).

Remark 5.4. The implication(D(X) ≃ limD(Xi)

)⇒

(Dperf(X) ≃ limDperf(Xi)

)in Proposition5.1

cannot be reversed. For example, considerX = Spec(Zp) andXi := Spec(Z/pn). ThenDperf(X) ≃limDperf(Xi) as in Theorem4.1. On the other hand,D(X) 6= limD(Xi): theZp-moduleQp maps to0 ineachD(Xi), and hence also inlimD(Xi), soD(X) → limD(Xi) is not even faithful.

As a special case of Proposition5.1, one obtains the following description of formal projective space.

Example 5.5. Let A be a noetherian ring that isI-adically complete for an idealI. Let X be a properA-space. ThenX = colimXn in the category of qcqs spaces, whereXn = X ×Spec(A) Spec(A/I

n)is the (classical) displayed fibre product. Indeed, by Proposition 5.1, it is enough to showDperf(X) ≃limDperf(Xn), which is a consequence of formal GAGA. (For example, one canuse [Lur11c, Theorem5.3.2] and the observation that a pseudo-coherent complexK ∈ D(X) is perfect if and only ifK|X1 is so.)Moreover, by Remark4.6, if X is a scheme, thenX ≃ colimXn in the category of all schemes.

We can now explain how to generalize certain well-known “formal glueing” results, such as [BL95]. Webegin with the following criterion for establishing such glueing features in a general setting.

Proposition 5.6. Fix a qcqs algebraic spaceX equipped with a constructible closed subspaceZ ⊂ X. Letπ : Y → X be a qcqs map algebraic spaces such thatπ∗ induces an equivalenceDZ(X) ≃ Dπ−1(Z)(Y ).LetU = X \ Z, andV = Y \ π−1(Z). Then one has:

20

(1) The natural mapΦ : D(X) → D(Y )×D(V ) D(U) is an equivalence.(2) Φ inducesDperf(X) ≃ Dperf(Y )×Dperf(V ) Dperf(U).(3) Φ inducesVect(X) ≃ Vect(Y )×Vect(V ) Vect(U).(4) If π is flat, thenΦ inducesQCoh(X) ≃ QCoh(Y )×QCoh(V ) QCoh(U).(5) The fibre square

Vj

//

π��

Y

π��

Uj

// X

is a pushout in qcqs algebraic spaces.(6) All of the above are also true after flat qcqs base change onX.

Proof. Write objects inD(Y ) ×D(V ) D(U) as triples(K,L, η) whereK ∈ D(Y ), L ∈ D(U), andη :j∗K ≃ π∗L. We first check (1). For full faithfulness, we may work locally onX. As all pushforward andpullback functors involved are cocontinuous, by the projection formula, we reduce to checking that

OXa→ π∗OY ⊕ j∗OU

b→ π∗j∗OV (1)

is a cofibre sequence. This can be checked after applying the conservative functorΓZ(−)⊕(−⊗j∗OU

). The

latter follows from the following sequence of assertions: the mapΓZ(a) is an isomorphism, the mapΓZ(b)is 0, and the mapa⊗j∗OU is the inclusion of a direct summand with the projection on the complement givenby b ⊗ j∗OU . These result from base change in coherent cohomology, and the assumptionπ∗ : DZ(X) ≃Dπ−1(Z)(Y ). For essential surjectivity, there is a functorΨ : D(Y )×D(V )D(U) → D(X) given as follows:if g := (K,L, η) ∈ D(Y )×D(V ) D(U) as before, then set

Ψ(g) := fib(π∗K ⊕ j∗L→ (π ◦ j)∗j

∗K),

where the map is induced byη. Note by construction thatΨ is the right adjoint6 to Φ. We will check thatΨ ◦ Φ ≃ id andΦ ◦ Ψ ≃ id via the (co)units of the adjunction. The first assertion is automatic from thefull faithfulness ofΦ. For the second, note thatj∗Ψ(g) = L, while ΓZ(Ψ(g)) ≃ ΓZ(π∗K) corresponds toΓπ−1(Z)(K) under the equivalenceDπ−1(Z)(Y ) ≃ DZ(X) induced byπ. Now for (K,L, η) as above, onehas a canonical cofibre sequence

Γπ−1(Z)(K) → K → j∗L

coming fromη. One then checksΦ(Ψ(g)) → g induces isomorphisms after projection toD(Y ) (as itinduces isomorphisms after applyingΓπ−1(Z)(−) andj∗) andD(U) separately, and thusΦ ◦Ψ ≃ id.

To get (2), we repeat the first half of the proof of Proposition5.1.For (3), we argue as in (2) using that the essential image of the fully faithful symmetric monoidal functor

Vect(S) → D(S) is exactly the set of dualizable objects inD≤0perf(S) for any qcqs spaceS by [Lur11d,

Corollary 2.7.33]. In order to apply this, we must check thatK ∈ Dperf(X) is connective if and only ifπ∗K andj∗K are connective. The forward direction is clear. For the converse, as connectivity of perfectcomplexes can be detected are restriction to a stratification, it is enough to check thatπ : Y → X issurjective overZ; if this was false, thenπ∗ : DZ(X) → Dπ−1(Z)(Y ) would have a non-trivial kernel (givenby the structure sheaf of the residue field at the missing point), which contradicts the assumption.

For (4), the flatness ofπ showsΦ restricts to a functorQCoh(X) → QCoh(Y ) ×QCoh(V ) QCoh(U).Thanks to (1), it is now enough to check: forK ∈ D(X), if j∗K andπ∗K are quasi-coherent sheaves, soisK. For suchK, we knowHi(K) ∈ DZ(X) for i 6= 0 asj∗K is a sheaf. Asπ is flat andπ∗K is a sheaf,

6The existence of the right adjoint can also be seen as follows: [Lur09, Proposition 5.5.3.13] and [Lur11b, Theorem 1.1.4.4]show that the fibre productD(Y ) ×D(V ) D(U) is a presentable stable∞-category, [Lur09, Proposition 5.5.3.12] shows thatΦpreserves colimits, whence [Lur09, Corollary 5.5.2.9] gives the right adjoint.

21

it follows thatπ∗Hi(K) = Hi(π∗K) = 0 for i 6= 0, and thusHi(K) = 0 for i 6= 0 by the assumptionπ∗ : DZ(X) ≃ Dπ−1(Z)(X), soK ≃ H0(K) is a sheaf.

Lastly, (5) follows from Proposition5.1and (2) (or (1)), while (6) comes from Remark5.2. �

Remark 5.7. In Proposition5.6, it is important to work with∞-categories instead of their1-categoricaltruncations as the formation of the homotopy-category is incompatible with fibre products. More concretely,the natural mapDcl(X) → Dcl(Y ) ×Dcl(V ) D

cl(U) is essentially surjective and full (which follows fromthe∞-categorical assertion for formal reasons), but can often fail to be faithful. Intuitively, this happensasDcl(Y ) ×Dcl(V ) D

cl(U) forgets “how” objects overY andU are being identified overV . An explicitexample illustrating this failure is given in Example5.15.

Proposition5.6 specializes to a few commonly encountered geometric situations. To illustrate these, letπ : Y → X be a qcqs map of qcqs algebraic spaces, and fix a constructibleclosed subspaceZ ⊂ X. ThehypothesisDZ(X) ≃ Dπ−1(Z)(Y ) from Proposition5.6is satisfied (and consequently the conclusions thereapply) notably in the following examples:

Example 5.8. The mapπ is flat, and an isomorphism overZ. For example,π could be an etale neighbour-hood ofZ in X, or one could takeX to be a noetherian affine withY the completion ofX alongZ (inthe sense of ring theory). The assumptionDZ(X) ≃ Dπ−1(Z)(Y ) for suchπ comes from Lemma5.12(2)below. The consequenceD(X) ≃ D(Y )×D(V )D(U) may be viewed as a derived variant of formal glueingresults due to Artin [Art70, §2] (which pertains to noetherian case) and Ferrand-Raynaud[FR70] (which isliterally Proposition5.6(4)); see also [Sta14, Tag 05ER] for more references.

Example 5.9. The spaceX is a separated smooth scheme of relative dimensiond over some base ringR,Z ⊂ X is the image of a sections : Spec(R) → X, andY = Spec(limAn), whereAn = Γ(Zn,OZn) is thering of functions on then-fold infinitesimal thickening onZ in X. The assumptionDZ(X) ≃ Dπ−1(Z)(Y )comes from Lemma5.12 (2) (or (1)) below. Zariski locally onSpec(R), one may choose a local co-ordinates definingZ ⊂ X, soA := limAn ≃ RJt1, . . . , tdK andV = Spec(A) \ Spec(A/(t1, . . . , td)).The consequenceVect(X) ≃ Vect(Y ) ×Vect(V ) Vect(U), in the special case whered = 1, recovers theBeauville-Laszlo theorem [BL95, §4, Example] in the form in which it is often used. Note that this is notcovered by Example5.8as completions often fail to be flat in the non-noetherian case. In fact, ifR is not acoherent ring, thenA ≃ RJtK can fail to be evenR-flat; see [Cha60].

Example 5.10. The spaceX = Spec(A) is an affine scheme for some ringA, the closed subspaceZ ⊂

X is cut out by a regular elementt ∈ A, andY = Spec(A), whereA = limA/tn. The assumptionDZ(X) ≃ Dπ−1(Z)(Y ) comes from Lemma5.12 (2) (or (1)) below. This case recovers the Beauville-Laszlo equivalence [BL95, §3, Theorem]

Modt(A) ≃ Modt(A)×Mod(A[ 1t])Mod(A[

1

t]),

as explained in Corollary5.14; hereModt(−) ⊂ Mod(−) is the full subcategory of allt-regular modules.Note further that this equivalence does not extend to allA-modules; see Example5.15. Proposition5.6shows that such an equivalence can be salvaged at the derivedlevel, i.e., the failure of the classical statementfor modules is the cost of ignoringTor groups.

Remark 5.11. It is commonly asserted that the glueing result discussed inExample5.8 is a direct conse-quence of faithfully flat descent for the coveringg : Y ⊔ U → X. However, this is not clear to us: thelatter statement would realizeVect(X) as the equalizer of the two evident mapsVect(Y ) × Vect(U) →Vect(Y ×X Y ) × Vect(V ) × Vect(U), which entails recording an isomorphism onY ×X Y (and higherfibre products, ifVect(−) is replaced byD(−)) as part of the descent data. It is nevertheless aconsequenceof the formal glueing result that this additional data is extraneous.

The next lemma was used above.22

Lemma 5.12. Letπ : Y → X be a map of qcqs algebraic spaces. Fix a finitely presented closed subspaceZ ⊂ X. Assume one the following:

(1) π is quasi-affine, and there exists aK ∈ Dperf(X) with supportZ such thatK ≃ π∗π∗K.

(2) π is an isomorphism overZ in the derived sense, i.e.,Z ×LX Y ≃ Z.

Thenπ∗ inducesDZ(X) ≃ Dπ−1(Z)(Y ).

Proof. First consider (1). The claim is local, so we may assumeX is affine, andY is quasi-affine. Inthis case, by Thomason’s [Tho97, Lemma 3.14] and approximation by perfect complexes with supportconstraints, the smallest stable subcategory〈K〉 ofD(X) containingK and closed under colimits is exactlyDZ(X), and similarly〈L〉 = Dπ−1(Z)(Y ) for L = π∗K; here we use that any stable subcategory ofD(X)closed under colimits is automatically an ideal (as〈OX〉 = D(X) and〈OY 〉 = D(Y ), sinceX andY arequasi-affine). As bothπ∗ andπ∗ are cocontinuous, it follows thatπ∗ ◦π∗ ≃ id onDZ(X), andπ∗ ◦π∗ ≃ idonDπ−1(Z)(Y ) as the same is true on generators by base change in coherent cohomology.

Now consider (2). We first check thatπ∗π∗K ≃ K for K ∈ DZ(X). By cocontinuity, one may assumeK is compact. By filteringK suitably, one reduces to the case whereK comes from anOZ -complex(but is not necessarily compact inD(X) any more). The claim now follows by base change in coherentcohomology and the assumption onZ. It remains to check thatL ≃ π∗π∗L for L ∈ Dπ−1(Z)(Y ). If Lcomes from anOπ−1(Z)-complex, then this follows from base change in coherent cohomology. One thenreduces to this case by cocontinuity and compact generationof Dπ−1(Z)(Y ). �

Remark 5.13. Lemma5.12(2) is closely related to [TT90, Theorem 2.6.3]; the latter imposes a strongerflatness constraint. Note also that the hypothesis of finite presentation onZ is necessary. Indeed, otherwisewe may takeX = Spec(colimCJt

1n K), andY = Z = Spec(C) with the mapt

1n 7→ 0 for all n.

We also explain why Proposition5.6recovers the classical Beauville-Laszlo theorem.

Corollary 5.14. Fix notation as in Example5.10. The base change functor gives an equivalence

φ : Modt(A) ≃ Modt(A)×Mod(A[ 1t]) Mod(A[

1

t]).

Proof. The mapφ has a right adjointψ given by(M,N, η) 7→ ker(M ⊕N →M [1t ]) with evident notation.We first check thatψ ◦ φ ≃ id. For this, fix someM ∈ Modt(A). Tensoring the resulting exact sequence

0 →M →M [1

t] →M [

1

t]/M → 0

with A, as− ⊗LA A is the identity ont∞-torsion modules, one concludes thatM → M [1t ] induces an

isomorphism onπi(−⊗LA A) for i > 0. The proof of Proposition5.6(1) then shows that

0 →M →(M ⊗A A

)⊕M [

1

t] →M ⊗A A[

1

t] → 0

is an exact sequence, soψ ◦ φ ≃ id, and thusφ is fully faithful. For essential surjectivity, fix someM ∈

Modt(A),N ∈ Mod(A[1t ]), and an isomorphismη :M [1t ] ≃ N ⊗A A. DefineK := M ×M [ 1t]

(N ⊗L

A A)

as a fibre product inD(A). Then projection inducesµ : K[1t ] ≃ N ⊗LA A, and thus an identification

fib(K → M) ≃ fib(N ⊗LA A → N ⊗A A). In particular, this fibre is connected and uniquelyt-divisible.

The triple(K,N, µ) defines an object ofD(A)×D(A[ 1t])D(A[1t ]), and thus comes from a uniqueL ∈ D(A)

by Proposition5.6(1). AsL[1t ] ≃ N , we know thatH i(L) is t∞-torsion fori 6= 0. We also have

L⊗LA A/(t) ≃ L⊗L

A A⊗LAA/(t) ≃ K ⊗L

AA/(t) ≃M/(t),

where the last equality uses thatM is t-regular, and thatK →M had a uniquelyt-divisible fibre. It followsthatH i(L) is uniquelyt-divisible for i 6= 0, 1, H0(L) is t-regular, andH1(L) is t-divisible. The previous

23

reduction then showsH i(L) ≃ 0 for i 6= 0, 1. As−⊗LA A is the identity operation ont∞-torsion modules,

we getH1(L) ≃ H1(L) ⊗LA A ≃ H1(L ⊗L

A A) = H1(K) = 0. Thus,L ≃ H0(L) is at-regular module,and one checks thatφ(L) = (M,N, η), proving the claim. �

The proof of Proposition5.6 takes place in the derived category. This is necessary: the equivalenceΦin Proposition5.6does not induce an equivalence on theabeliancategories of quasi-coherent sheaves. Anexample illustrating this failure (coming fromC∞-function theory) is mentioned in [BL95, §4, Remark 4],so we recall a different one (coming from rigid analytic geometry) here for the convenience of the reader.

Example 5.15. Fix a primep, and letA be the ring of germs of bounded algebraic functions at0 on thep-adic unit disc, i.e.,

A = colim(Zp[x] → Zp[

x

p] → Zp[

x

p2] → . . .

).

Note that bothp andx are regular elements ofA. In fact,A is a domain: we may viewA as the subring ofQp[x] spanned by polynomialsf(x) with f(0) ∈ Zp. Thusx is uniquelyp-divisible inA by construction,soA/pn ≃ Z/pn, and thusA = Zp (where the completion isp-adic). SetX = Spec(A), Y = Spec(A),Z = Spec(A/p) with U andV as in Proposition5.6. Now if we considerM = A/(x), then the map

η :M →(M ⊗A A

)⊕M [

1

p]

has a non-trivial kernelK: the elements0 6= xpn ∈ M for all n ≥ 1 span a copy ofQp/Zp in K. As

M ⊗A A ≃ A/(x) ≃ Zp andM [1p ] are bothp-torsion free, studyingHom(A,M) then shows that

QCoh(X) → QCoh(Y )×QCoh(V ) QCoh(U)

is not faithful. This failure is explained by the derived picture as follows: the sequence

M →(M ⊗L

A A)⊕M [

1

p] →M ⊗L

A A[1

p]

is a cofibre sequence inD(X) ≃ D(A) by Proposition5.6(1), but the sequence ofA-modules

0 →Mη→

(M ⊗A A

)⊕M [

1

p] →M ⊗A A[

1

p]

obtained by applyingπ0(−) to the above cofibre sequence is not exact on the left. In fact,one computes

M ⊗LA A ≃ cofib(A

x→ A)⊗L

A A ≃ cofib(Ax→ A) ≃ A⊕ A[1] ≃ Zp ⊕ Zp[1],

where the second-to-last equality uses thatx = 0 on A. This gives

M ⊗LA A[

1

p] ≃ A[

1

p]⊕ A[

1

p][1] ≃ Qp ⊕Qp[1]

by invertingp, and thus the kernelK = ker(η) above is identified as

K ≃ coker(π1(M ⊗L

A A) → π1(M ⊗LA A[

1

p]))≃ A[

1

p]/A ≃ Qp/Zp.

Note further that this example also shows that Proposition5.6 is not true for classical derived categories.Indeed, writeΦcl : Dcl(X) → Dcl(Y )×Dcl(V ) D

cl(U) for the obvious functor. Then one computes

Hom(OX , M) ≃M

in Dcl(X), whileHom(Φcl(OX),Φcl(M )) ≃M/K

in Dcl(Y )×Dcl(V ) Dcl(U). Of course, the latter is entirely a consequence of the non-faithfulness of

h(D(Y )×D(V ) D(U)

)→ Dcl(Y )×Dcl(V ) D

cl(U).

24

6. ALGEBRAIZATION OF PRODUCTS: SCHEMES

The goal of this section is to prove the following theorem.

Theorem 6.1. Fix a setI of rings{Ai}i∈I with productA :=∏

iAi, and a qcqs schemeX. ThenX(A) ≃∏iX(Ai) via the natural map.

The result is sharp, as illustrated in Example8.5.

Remark 6.2. We will prove a version of Theorem6.1 for qcqs algebraic spaces in§7. In fact, the proofsare also similar. The main difference is that the case of spaces relies on Theorem2.1, while, in the world ofschemes, we may use Theorem1.7. A practical consequence is that the proof for schemes is considerablymore elementary (but not any more direct) than the proof for spaces.

Fix {Ai}i∈I andX as in the theorem. For proving injectivity ofX(A) →∏

iX(Ai), we use derivedcategory techniques (as these shall be handy later), thoughone can also do this directly. The first step is toidentify perfect complexes onA as products of (certain) perfect complexes on eachAi. More precisely:

Lemma 6.3. The evident mapφ : Dperf(A) →∏

iDperf(Ai) is fully faithful, andK ≃∏

iK ⊗A Ai foranyK ∈ Dperf(A).

Proof. AsDperf(A) is generated byA under finite colimits and retracts, for the first claim, it is enough tocheck thatA ≃ Hom(φ(A), φ(A)), which is clear. The second claim is proven similarly. �

Next, we prove a special case of Theorem6.1:

Lemma 6.4. If X is quasi-affine, thenX(A) ≃∏

iX(Ai).

Proof. The assertionX(A) ≃∏

iX(Ai) is clear ifX is affine. Hence, ifX is quasi-affine, then oneimmediately obtains injectivity ofX(A) →

∏iX(Ai). For surjectivity, fix an affineY containingX as an

open subscheme, with constructible closed complementZ. We must show that ifa : Spec(A) → Y factorsthroughX over eachSpec(Ai) ⊂ Spec(A), thena factors throughX, or, equivalently, thata∗OZ = 0.Choose someK ∈ DZ(Y ) ∩ Dperf(Y ). Thena∗K =

∏i(a

∗K) ⊗A Ai ≃ 0. As OZ can be written as afiltered colimit of suchK ’s, one finds thata∗OZ = 0, soa−1(Z) = ∅, as wanted. �

Using this special case, we can establish injectivity:

Lemma 6.5. The mapX(A) →∏

iX(Ai) is injective.

Proof. Fix two mapsa, b : Spec(A) → X which induce the same mapai = bi overSpec(Ai) ⊂ Spec(A).Now consider the pullbackz : Z → Spec(A) of ∆ : X → X ×X along(a, b) : Spec(A) → X ×X. ThenZ → Spec(A) is a quasi-compact monomorphism as∆ is so. In particular,Z is quasi-affine. Moreover,Zadmits sections over eachSpec(Ai) ⊂ Spec(A). Lemma6.4 gives a unique mapSpec(A) → Z inducingthe given sections over eachSpec(Ai). It follows thatZ → Spec(A) is a monomorphism with a section,and thus an isomorphism. This immediately givesa = b, as wanted. �

Remark 6.6. The proof of Lemma6.5appliesmutatis mutandisto qcqs algebraic spaces.

We now come to the interesting part: the surjectivity ofX(A) →∏

iX(Ai). Fix mapsfi : Spec(Ai) →X. We do not know how to directly construct a mapf : Spec(A) → X extending eachfi. Instead, we firstdefine a functorG : QCoh(X) → ModA via

G(F) :=∏

i

Γ(Spec(Ai), f∗i F).

The functorG will not be the pullback functor for the desired mapf : Spec(A) → X. In fact,G does notpreserve (infinite) direct sums unlessI is finite, soG cannot be a pullback. However, we will later build anew functorF (which will be the desired pullback) fromG, using crucially the following fact:

25

Lemma 6.7. G preserves locally finitely presented objects.

Proof. Choose affine open covers{U1, . . . , Ur} of X and{V1, . . . , Vm} of Spec(Ai) such thatfi carrieseachVk to someUj , and the numbersr andm are bounded independently ofi ∈ I; this is possible byLemma6.9. We name the index setsJ = {1, . . . , r} andK = {1, . . . ,m} for notational simplicity. Choosea locally finitely presentedF ∈ QCoh(X), and writeM := G(F) ∈ ModA. We must check thatM isfinitely presented. For eachj ∈ J , pick a presentation

O⊕ℓjUj

Aj→ O

⊕nj

Uj

Bj→ F|Uj

→ 0.

Setℓ = max(ℓj) andn = max(nj); these are “absolute” constants depending only onX andF. Fix someindex i ∈ I, and letMi := Γ(Spec(Ai), f

∗i F). ThenMi is a finiteAi-module, andMi|Vk

is generatedby ≤ n sections. Lemma6.8 shows thatMi is itself generated byn · m sections. This gives a surjection

A⊕n·mi

Qi→Mi. Note thatn andm are independent of the choseni ∈ I. Taking products, we get a surjective

mapA⊕n·m Q→M , which shows thatM is finitely generated.

Let Ki = ker(Qi) ⊂ A⊕n·mi , andK = ker(Q) ⊂ A⊕n·m. As K =

∏iKi, we must show thatKi

is generated byn′ elements, for somen′ independent ofi. We will do so by bounding the number ofgenerators for its restriction to eachVk. Fix somek ∈ K, and pickj ∈ J such thatfi(Vk) ⊂ Uj. SetLk := ker(f∗i Bj|Vk

). Then there is a short exact sequence

1 → Lk → O⊕nj

Vk→ Mi|Vk

→ 1.

On the other hand, we also have a short exact sequence

1 → Ki|Vk→ O

⊕n·mVk

→ Mi|Vk→ 1

by definition ofKi. Taking a fibre product of the two penultimate maps in these exact sequences, and usingthatExt1Vk

(OVk,−) = 0 asVk is affine, one obtains (non-canonically) an isomorphism

O⊕n·mVk

⊕ Lk ≃ O⊕nj

Vk⊕ Ki|Vk

.

Note thatLk is generated by≤ ℓ global sections asf∗i (Aj)|Vkfactors as

O⊕ℓjVk

։ Lk → O⊕nj

Vk,

where the first map is surjective and the second is injective (and recall:ℓ = max(ℓj)). It follows thatKi|Vk

is generated≤ N := n ·m+ ℓ sections; note thatN is independent ofi. Another application of Lemma6.8shows thatKi is generated≤ n′ := N ·m elements, as wanted. �

The following two elementary results were used above. The first bounds the number of generators of amodule in terms of local data.

Lemma 6.8. Fix a ring R, and a finiteR-moduleM . Assume there exists an integern ≥ 0 and an opencover{U1, . . . , Um} of Spec(R) such thatM |Ui

is generated by≤ n sections. ThenM is generated by≤ n ·m elements.

The proof below was explained to me by de Jong.

Proof. SetY = Spec(R). We work by induction onm. There is nothing to prove ifm = 1, so assumem ≥ 2. LetZ := Y \Um, viewed as a reduced closed subscheme ofY . ThenZ is an affine scheme coveredby {V1, . . . , Vm−1}, whereVi = Ui ∩ Z. By the inductive hypotheses,M |Z is generated by≤ n · (m− 1)

sections. Lifting these sections, we can find a mapR⊕n·(m−1) → M that is surjective on some openneighbourhoodV of Z (by Nakayama’s lemma). ThenZ ′ := Y \ V is a closed subscheme ofY , andM |Z′

is generated byn sections (asZ ′ ⊂ Um). Lifting a generating set, and adding to the generators found earlier,26

we obtain a mapR⊕n·m ≃ R⊕n⊕R⊕n·(m−1) →M . This map is surjective overV ⊔Z ′ = Spec(R), whichproves the claim. �

The second bounds the number of affines needed to cover a quasi-affine scheme, universally.

Lemma 6.9. Let j : W → Y be a quasi-affine morphism of qcqs schemes. Then there existsan integermsuch that: for any mapSpec(B) → Y , the pullbackW ×Y Spec(B) is covered bym affines.

Proof. By replacingY with the affineY -schemeSpecY(j∗OW ), we may assumej is a quasi-compact

open immersion. LetZ = Y \ W , and choose a locally finitely generated ideal sheafI ⊂ OY definingZ. Choose an affine open cover{U1, . . . , Uk} of Y such thatI|Ui

is generated≤ r global sections (forsomer), and setm = k · r. Fix some mapj : Spec(B) → Y from an affine scheme. Lemma6.8then gives sectionsf1, . . . , fm ∈ Γ(Spec(B), j∗I) generatingj∗I. The corresponding distinguished opens{D(f1), . . . ,D(fm)} of Spec(B) give the desired open cover forW×Y Spec(B) = Spec(B)\j−1(Z). �

Write QCohfp(X) ⊂ QCoh(X) for the full subcategory of finitely presented quasi-coherent sheaves,and similarly forModfp,A ⊂ ModA. Lemma6.7shows thatG restricts to a functorGfp : QCohfp(X) →Modfp(A). This functor has desirable properties:

Lemma 6.10.Gfp is symmetric monoidal and preserves finite colimits.

Proof. ForF1,F2 ∈ QCoh(X), there is a natural mapG(F1)⊗G(F2) → G(F1⊗F2); we will first show thismap is an isomorphism ifFi ∈ QCohfp(X). For this, note thatModfp,A →

∏iModfp,Ai

is fully faithfuland symmetric monoidal. Indeed, the latter is automatic, while the former is a consequence ofHomA(M,−)commuting with flat base change onA forM ∈ Modfp,A. Thus, the assertions forGfp can be checked aftercomposing with the projectionModfp,A → Modfp,Ai

. But the resulting functorQCoh(X) → Modfp,Ai

is simplyΓ(Spec(Ai), f∗i (−)), which is clearly symmetric monoidal. The preservation of finite colimits

is proven similarly asModfp,A →∏

iModfp,Aipreserves finite colimits, and because finite colimits are

computed “termwise” in the target. �

To build the promised functorF , we use a result of Deligne [Har66, Appendix, Proposition 2] to identifyQCoh(X) in terms ofQCohfp(X).

Lemma 6.11(Deligne). The natural inclusionQCohfp(X) ⊂ QCoh(X) extends to a symmetric monoidalequivalenceInd(QCohfp(X)) = QCoh(X) given by{Ai} 7→ colimAi.

We can now prove Theorem1.3by applying Theorem1.7.

Proof of Theorem1.3. The symmetric monoidal functorGfp : QCohfp(X) → Modfp,A defines a symmet-ric monoidal functorF : QCoh(X) → ModA by passage to ind-completions. Moreover, bothQCohfp(X)and Modfp,A have finite colimits andGfp preserves these. By formal nonsense,F preserves all col-imits. Hence, by Theorem1.7, there is a unique mapf : Spec(A) → X such thatF = f∗. Note

that the compositionQCoh(X)F→ ModA → ModAi

is identified withΓ(Spec(Ai), f∗i (−)) as both

are cocontinuous and agree on the compact objectsQCohfp(X) ⊂ QCoh(X). Hence, the composition

Spec(Ai) ⊂ Spec(A)f→ X inducesf∗i on pullback, and must therefore coincide withfi by Theorem1.7.

It follows thatf is the desired extension. �

27

7. ALGEBRAIZATION OF PRODUCTS: ALGEBRAIC SPACES

The goal of this section is to prove the following theorem.

Theorem 7.1. Fix a setI of rings{Ai}i∈I with productA :=∏

iAi, and a qcqs algebraic spaceX. ThenX(A) ≃

∏iX(Ai) via the natural map.

The injectivity of the map appearing in Theorem7.1is proven as in Theorem6.1. For surjectivity, we useTheorem2.1. To apply this theorem, one must construct a symmetric monoidal functorF : Dperf(X) →Dperf(

∏iAi) starting from a family of maps{fi : Spec(Ai) → X}i∈I . In analogy with the proof of

Theorem6.1, the obvious guess is to useF (K) =∏

i Γ(Spec(Ai), f∗i K) ∈ D(A). In fact, if F = f∗

for some mapf : Spec(A) → X, thenF is forced to be given by this formula7. However, in general,it is not clear whyF (K) thus defined is a perfect complex: an arbitrary product of perfect complexesKi ∈ Dperf(Ai) is typically notA-perfect. Indeed, various numerical invariants associated to theKi’s (suchas the cohomological amplitude, the Tor amplitude, the minimal number of generators forH0(Ki), etc.)might be unbounded asi varies, which immediately precludes

∏iKi from beingA-perfect; see Example

8.3 for an explicit example. In§7.1 and§7.2, we show that the numerical obstruction is the only one: iftheKi’s are presented by a “bounded amount of data” (asi varies), then

∏iKi is indeed perfect. The

phrase “bounded amount of data” is made precise by bounding the number and embedding dimension of theprojective modules occurring in a presentation forKi over a Nisnevich cover8 of Spec(Ai) of bounded size;the key result, Lemma7.11, is that a local bound ofN ∈ N implies a global bound off(N) ∈ N for somefunctionf : N → N (which, crucially, is independent of the ring under consideration). It is then relativelystraightforward to check that the functorF defined above does the job, as we do in§7.3.

7.1. Bounding Nisnevich covers.All algebraic spaces appearing in this subsection are assumed to be qcqs.Recall that a mapf : U → Y of algebraic spaces is called aNisnevich coverif it is an etale cover that admitssections over a constructible stratification ofY (see [Lur11a, §1]). We will need a quantitative variant:

Definition 7.2. A Nisnevich coverf : U → Y of an algebraic spaceY haslength≤ m if there exists a flag∅ = Z0 ⊂ Z1 ⊂ · · · ⊂ Zm := Y of constructible closed subsets such thatf |Zi\Zi−1

has a section for alli.

Note that every Nisnevich coverf : U → Y has≤ m for somem > 0. Moreover, this property is stableunder base change.

Lemma 7.3. Fix a Nisnevich coverf : U → Y of an algebraic spaceY of length≤ m. For any mapg : Y ′ → Y , the base changeU ×Y Y

′ → Y ′ of f alongg is a Nisnevich cover of length≤ m.

Proof. As Y ′ andY are qcqs, the mapg is a qcqs map, and thusg−1 preserves constructibility, whichimmediately gives the lemma. �

Lengths multiply under compositions.

Lemma 7.4. Fix a compositeY1g1→ Y2

g2→ Y3 of Nisnevich covers of algebraic spaces such thatgi has

length≤ mi for i = {1, 2} and suitablemi ∈ N. Theng2 ◦ g1 has length≤ m1 ·m2.

Proof. Let Z• := {∅ = Z0 ⊂ Z1 ⊂ · · · ⊂ Zm2 = Y3} andW• := {∅ = W0 ⊂ W1 ⊂ · · · ⊂ Wm1 = Y2}be flags of constructible closed subsets witnessing the lengths ofg2 andg1 respectively. We will inductivelyconstruct a lengthm2 flagKi

• of constructible closed subsetsKij ⊂ Zi such thatKi

j containsZi−1 for allj, andg2 ◦ g1 admits sections over eachKi

j \ Kij−1. Putting the variousKi

• together then gives a flag inY3 of sizem1 · m2 with the required properties. Fori = 1, we simply useK1

• := W• ∩ Z1, whereZ1

is viewed as a closed subset ofY2 via some chosen section ofg2 overZ1. Assume such flags have beenconstructed forZi. ThenZi+1 \ Zi may be viewed as a subscheme ofY2 via some chosen section. Thus,

7This is only true for perfect complexes, not “large” quasi-coherent complexes.8As we work with algebraic spaces, we are forced to use Nisnevich covers instead of Zariski ones.

28

Ki+1,′• := W• ∩ (Zi+1 \ Zi) defines a flag of constructible closed subsets ofZi+1 \ Zi of lengthm2 such

thatg2 ◦ g1 admits sections overKi+1,′

j \Ki+1,′

j−1 . SettingKi+1• = Ki+1,′ ∪ Zi gives the desired flag. �

As a result, lengths behave predictably under Zariski covers.

Example 7.5. Let f : U → Y be a Nisnevich cover of algebraic spaces with length≤ m. Fix an open cover{U1, . . . , Uk} of U . Then the composite⊔Ui → Y is a Nisnevich cover of length≤ m · k. To see this, it isenough to check thatg : ⊔Ui → U is a Nisnevich cover of length≤ k. We show this by induction onk. Ifk = 1, the claim is clear. In general, setZk := U andZk−1 := U \ U1. Note that⊔Ui → U has a sectionoverU1 := Zk \ Zk−1. AsU1 ∩ Zk−1 = ∅, the inductive hypotheses applies to the restriction ofg toZk−1

to give a flag∅ = Z0 ⊂ Z1 ⊂ · · · ⊂ Zk−1 of closed subschemes such thatg admits sections overZi \Zi−1.It is then clear thatg has length≤ k.

Using lengths, one can bound the minimal number of generators of a module over a ring in terms of thecorresponding number over a Nisnevich cover, in analogy with Lemma6.8.

Lemma 7.6. Fix a Nisnevich coverf : U → Y of length≤ m of an affine schemeY = Spec(R), and someM ∈ ModR. If f∗M is generated by≤ N global sections, thenM is generated by≤ Nm global sections.

Proof. We prove the claim by induction onm. Choose a flag∅ = Z0 ⊂ Z1 ⊂ . . . Zm := Y of constructibleclosed subsets ofY such thatf admits sections overZi \ Zi−1. If m = 1, thenZ1 = Y , so there is nothingto show. Form > 1, choose a mapφ : R⊕m → M that is surjective overZ1; this is always possible asM |Z1 is generated by≤ m sections by the assumption onf , and becauseZ andX are affine. The cokernelQ = coker(φ) is a finitely presentedR-module whose support is a closed subschemeW := Spec(R/I) ⊂Spec(R) that does not intersectZ1. The restrictionf |W is then a Nisnevich cover of length≤ m − 1. AsQ is a quotient ofM/IM , one finds, by induction, a surjection(R/I)⊕N ·(m−1) → Q. Lifting sections toRandM gives a mapψ : R⊕N ·(m−1) →M whose image surjects ontoQ. The sumφ⊕ ψ : R⊕N ·m →M isthen easily seen to be surjective, proving the claim. �

One also has an analogue of Lemma6.9.

Lemma 7.7. Let j : W → Y be a quasi-affine morphism of algebraic spaces. Then there exists an integerm such that: for any mapSpec(B) → Y , the pullbackW ×Y Spec(B) is covered bym affines.

Proof. By replacingY with the affineY -spaceSpecY(j∗OW ), we may assumej is a quasi-compact open

immersion. LetZ = Y \W , and choose a locally finitely generated ideal sheafI ⊂ OY definingZ. Choosea Nisnevich coverf : U → Y of length≤ k such thatf∗I is generated≤ r global sections. Setm = k · r.Fix some mapj : Spec(B) → Y from an affine scheme. ThenW ×Y Spec(B) → Spec(B) is a quasi-compact open immersion defined by the idealj∗I. Lemma7.6, applied to the pullback off alongj, showsthat j∗I is defined bym global sectionsf1, . . . , fm ∈ Γ(Spec(B), j∗I). The corresponding distinguishedopens{D(f1), . . . ,D(fm)} of Spec(B) give the desired open cover forW ×Y Spec(B). �

We will actually need a more precise version of a special caseof Lemma7.7:

Lemma 7.8. Letf : U → Y be a Nisnevich cover of an algebraic spaceY of length≤ m′. There exists anintegerm such that: for any mapSpec(B) → Y , there exists a Nisnevich coverV → Spec(B) of length≤ m with V affine, and aY -mapV → U .

Proof. Lemma7.7applied tof gives an integerm′′ such thatV ′ := U ×Y Spec(B) admits an affine opencover{V ′

1 , . . . , V′m′′}. SetV = ⊔V ′

i , so the mapV → V ′ is a Nisnevich cover of length≤ m′′. Settingm = m′ ·m′′ then solves the problem. �

29

7.2. Bounding perfect complexes.The goal of this subsection is to formulate and prove the promisedresult bounding the size of the presentation of a perfect complex over an affine scheme in terms of the samedata over a bounded Nisnevich cover. We make the followingad hocdefinitions for the “size”:

Definition 7.9. Fix a ringR, an objectK ∈ Dperf(R), and some positive integerN . We say:(1) K locally has size≤ N if there exists a Nisnevich coverf : U → Spec(R) of length≤ N with

U affine such thatf∗K is represented by a finite complexP • of finite locally freeOU -modulesP i

with P i = 0 for |i| > N and eachP i being a retract ofO⊕NU .

(2) K globally has size≤ N if K can be represented as a complexP • of finite projectiveR-modulesP i with P i = 0 for |i| > N and eachP i being a retract ofR⊕N .

We record an elementary property of these notions.

Lemma 7.10. For any ringR, eachK ∈ Dperf(R) globally has size≤ N for someN > 0. Moreover, inthis case,K locally has size≤ N .

Proof. Clear from the definition. �

The key result is a converse to the previous lemma: one may propagate local bounds to global ones.

Lemma 7.11. There exists a functionf : N → N such that: for every ringR andK ∈ Dperf(R), if Klocally has size≤ N , thenK globally has size≤ f(N).

The proof below constructs anf with f ∼ O(N24N ); is it optimal?

Proof. The proof involves a nested induction. The “outer” induction is along the cohomological amplitude,while the “inner” induction is along the Tor amplitude. As both these quantities are bounded by twice thelocal size, both inductions are actually finite.

More precisely, we will recursively construct functionsfi : N → N for i = 0, . . . , 2N such that:

(1) For any ringR andK ∈ Dperf(R), if K locally has size≤ N and cohomological amplitude of size≤ i,thenK globally has size≤ fi(N).

One then definesf(N) = f2N (N). This clearly does the job because anyK ∈ Dperf(R) which locally hassizeN has cohomological amplitude of size≤ 2N .

Assume firsti = 0. We will recursively construct functionsf j0 : N → N for j = 0, . . . , 2N such that:

(a) For any ringR andK ∈ Dperf(R) with cohomological amplitude0, if K locally has size≤ N and Toramplitude of size≤ j, thenK globally has size≤ f j0 (N).

Takingf0 = f2N0 then solves the problem of constructingf0 as anyK ∈ Dperf(R) which locally has size≤ N also has Tor amplitude≤ 2N . Forj = 0, we have:

Claim 7.12. f00 (N) = N2 satisfies (b) forj = 0.

Proof of Claim. Fix someK ∈ Dperf(R) which locally has size≤ N (for someN ), and cohomologicaland Tor amplitude of size0. ThenK = M [k] for a finite projectiveR-moduleM and somek ∈ Z. Fix aNisnevich coverf : U → X of length≤ N with U affine, as well as a finite complexP • of finite locallyfreeOU -modules as in the definition of local size. Asτ>k(K) = 0, one checks (using thatU is affine) thatZk(P •) is a finite projectiveOU -module which occurs as a retract ofO

⊕NU . Moreover,Zk(P •) → Hk(f∗K)

is surjective by definition. Lemma7.6then shows thatM = Hk(K) is generated byN2 global sections. AsM is projective, one may realizeM as a summand ofR⊕N2

, which proves the claim. �

To construct the remainingf j0 inductively, fix some0 < j ≤ 2N and assume thatf j′

0 has been constructedfor j′ < j. It suffices to show: for any ringR and any non-zeroK ∈ Dperf(R) which locally has size≤ N ,cohomological amplitude0, and Tor amplitude≤ j, there exists a cofibre sequence

K ′ → R⊕N2[−k]

h→ K

30

wherek is the unique integer such thatHk(K) 6= 0, andh is surjective onHk. Indeed, thenK ′ ∈ Dperf(R)locally has size≤ N2 + N (by the formula for a mapping cone), cohomological amplitude 0, and Toramplitude≤ j − 1, so one may usef j0 (N) = N2 + f j−1

0 (N2 + N). The maph can be constructed usingthe technique from the previous paragraph, so we have constructedf0 satisfying (1).

We now constructfi for i > 0. Fix some0 < i ≤ 2N , and assume thatfj : N → N satisfying (1) havebeen constructed forj < i. We claim thatfi(N) = N2 + fi−1(N

2 + N) does the job. For this, fix someringR and0 6= K ∈ Dperf(R) with local size≤ N and cohomological amplitude of size≤ i. It is enoughto construct a cofibre sequence

R⊕N2[−k]

g→ K → Q

wherek is the largest integer withHk(K) 6= 0, andg is surjective onHk. Indeed, once such a cofibresequence is constructed,Q locally has size≤ N2 +N (by the formula for the mapping cone) and cohomo-logical amplitude of size≤ i − 1, soK globally has size≤ N2 + fi−1(N

2 + N) (by the formula for themapping cone). To construct this cofibre sequence, it is enough to constructg. Choosek as above. ThenHk(K) can be generated by≤ N2 elements by the argument used in the previous two paragraphs. Thisgives a surjective mapg : R⊕N2

→ Hk(K). Thencofib(K[−k] → Hk(K)) is 1-connected, so we may liftg to a mapg[−k] : R⊕N2

→ K[−k] that is surjective onH0; shifting byk gives the desired map. �

7.3. Proof of Theorem 7.1. Fix X, {Ai}, andA as in Theorem7.1. The proof of injectivity ofX(A) →∏iX(Ai) proceeds exactly as before. For surjectivity, we first show that a family of projective modules

over eachAi with uniformly bounded embedding dimension can be patched to a projective module overA.

Lemma 7.13. Fix projectiveAi-modulesPi. AssumePi is a retract ofA⊕Ni for someN independent ofi.

ThenP :=∏

i Pi is a projectiveA-module, and a retract ofA⊕N .

Proof. Choose projectorsǫi ∈ MN (Ai) realizingPi, i.e., ǫ2i = ǫi andPi = im(ǫi). Thenǫ =∏

i ǫi ∈MN (A) is a projector, andP = im(ǫ) is indeed projective; we use here that the formation of cokernelscommutes with arbitrary products in abelian groups. �

We can upgrade this to a patching result for perfect complexes.

Lemma 7.14. Fix Ki ∈ Dperf(Ai) which globally have size≤ N for someN independent ofi. ThenK :=

∏iKi ∈ Dperf(A).

Proof. We may represent eachKi by a finite complexP •i of finite projectiveAi-modules such thatPi = 0

for |i| > N andPi is a retract ofA⊕Ni . Lemma7.13shows thatP • :=

∏i P

•i ∈ D(A) has finite projective

terms withP i = 0 for |i| > N , soK ≃ P • is perfect. �

Using this patching result and the bounds from§7.2, we can finishing proving surjectivity.

Proof of Theorem7.1. We have already seen thatX(A) →∏

iX(Ai) is injective (see Remark6.6). For sur-jectivity, fix mapsfi : Spec(Ai) → X for i ∈ I. We claim that the associationK 7→

∏i Γ(Spec(Ai), f

∗i K)

defines a functorF ′ : D(X) → D(A) that preserves perfect complexes. Fix someK ∈ Dperf(A). ThenKlocally has size≤ N for someN ; here we implicitly use thatX has a Nisnevich cover by affine schemes(see [Sta14, Tag 08GL] or [Lur11c, Theorem 1.3.8]). Using Lemma7.8, one checks thatf∗i K locally hassize≤ N ′ for someN ′ independent ofi. By Lemma7.11, the complexf∗i K globally has size≤ N ′′ forsomeN ′′ independent ofi. By Lemma7.14, it follows thatF ′(K) is perfect. Using Lemma6.3, one easilychecks that the resulting functorF : Dperf(X) → Dperf(A) is symmetric monoidal. By Theorem2.1,one obtains a mapf : Spec(A) → X such thatf∗ = F . As the composition ofF with any projectionDperf(A) → Dperf(Ai) is simplyf∗i , it follows thatf extends eachfi, as wanted. �

31

8. SOME EXAMPLES AND APPLICATIONS

The main goal of this section is to record some special cases of Theorem6.1 and Theorem7.1 that canbe proven by hand. Along the way, we also give counterexamples illustrating the sharpness of these results.We begin with two examples where Theorem6.1can be proven directly.

Example 8.1. With notation as in Theorem6.1, assume that eachAi is local. Choose a Zariski cover{U1, . . . , Un} of X, and setU = ⊔jUi. Then every mapSpec(Ai) → X lifts to U , so the surjectivity ofX(A) →

∏iX(Ai) follows from that forU . The injectivity is proven as before.

Example 8.2. Let X = Pn, and fix a setI of rings{Ai}i∈I . WriteA =∏

iAi. We will showX(A) =∏iX(Ai) by interpretingX(R) as the collection of tuples(L, s0, . . . , sn) whereL ∈ Pic(R) andsi ∈ L

such thatR⊕n+1 si→ L is surjective (up to isomorphism). By suitably twisting, one first checks thatPic(A) →

∏i Pic(Ai) is fully faithful; herePic(−) denotes the Picardcategoryfunctor. It is then rel-

atively easy to see thatX(A) →∏

iX(Ai) is injective. For surjectivity, one must show: givenxi :=(Li, si,0, . . . , si,n) ∈ X(Ai), there existsx := (L, s0, . . . , sn) ∈ X(A) inducingxi. This follows by theargument in Lemma7.13. A similar argument works wheneverX is quasi-projective over an affine (usingthe trick from Lemma6.4).

The argument in Example8.2 (or, rather, Lemma7.13 ) crucially relies on bounding the embeddingdimension of the line bundles. This is necessary: an arbitrary product of line bundlesLi ∈ Pic(Ai) does notgive a line bundle onA, and thus Theorem6.1does not extend to Artin stacks, as the next example shows.

Example 8.3. LetX = BGm, and fix a set{Ai} of rings. ThenX(∏

iAi) 6=∏

iX(Ai) in general. Moreprecisely, the natural mapX(

∏iAi) →

∏iX(Ai) is not essentially surjective. To see this, it is enough

to exhibit rings a sequence{An} of rings with line bundlesMn ∈ Pic(An) such thatMn is generated byno fewer thanf(n) sections, wheref : N → N is an unbounded function; this is simply because any linebundle onSpec(

∏nAn) defines a line bundle onSpec(An) generated byN sections for someN ≫ 0

independent ofn. Such a sequence of line bundles{Mn} and rings{An} was constructed by Swan (withAn noetherian); see [Swa62].

The next example shows that Theorem6.1fails for the simplest Deligne-Mumford stacks; the underlyingreason is the purely topological fact that the classifying space of a finite group is infinite dimensional, thoughwe argue cohomologically in the algebraic context.

Example 8.4. Fix an algebraically closed fieldk of characteristic0, and letG be a non-trivial finite group.Let X = BG be the classifying stack ofG-torsors. We will construct affine schemesSpec(Ai) for eachi ∈ N, and mapsfi : Spec(Ai) → X such that⊔iSpec(Ai) → X does not factor through a mapSpec(

∏iAi) → X. For the construction, fix a primep dividing the order ofG, soH i

et(X,Z/p) 6= 0 for ar-bitrarily largei ∈ N9. For eachi ∈ N, choose an affine varietyUi := Spec(Ai) and a mapUi → X which isan isomorphism onH≤i

et (−,Z/p); this can always be done by looking at the quotient byG of the stabilizer-free locus in a sufficiently large representation ofG and using the Jounalou trick; see [Tot99, Lemma 1.6]or [MV99, §4.2]. Consider the resulting mapfi : Spec(Ai) → X. We claim that⊔ifi : ⊔Ui → Xdoes not factor through some mapf : Spec(

∏iAi) → X. Assume towards contradiction such anf does

exist. AsX is locally finitely presented, we can find a finitely presentedk-subalgebraA ⊂∏

iAi suchthat f factors through some mapα : Spec(A) → X. As Spec(A) is an algebraic variety, one knowsHk

et(Spec(A),Z/p) = 0 for k > dim(A) by Artin vanishing. In particular, it follows thatfi, viewed as the

composite mapSpec(Ai) → Spec(∏

iAi) → Spec(A) → X, induces the0 map onH>dim(A)et (−,Z/p) for

all i. However, fori ≫ 0, the mapfi induces a non-zero map onHk(−,Z/p) for somek > dim(A) byconstruction, which gives the desired contradiction.

9In fact, by [Qui71], one knows thatH∗

et(X,Z/p)/(nilpotents) is a finitely generatedZ/p-algebra of non-zero Krull dimen-sion, and thusHi

et(X,Z/p) cannot be0 for all i ≫ 0.32

The next example shows that Theorem6.1is false ifX is not qc or not qs.

Example 8.5. Take an infinite setI and setAi := k for some non-zero ringk. If X = ⊔iSpec(Ai) isthe displayed non-quasi-compact scheme, it is easy to see thatX(

∏iAi) →

∏iX(Ai) is not surjective.

Now setY to be the glueing ofX := Spec(∏

iAi) to itself along the identity onX ⊂ X. ThenY isquasi-compact, but not quasi-separated. It is easy to see that the two distinct mapsX → Y induce the samemapSpec(Ai) → Y , soY (

∏iAi) →

∏i Y (Ai) is not injective.

The next example contains a direct proof of an important special case of Theorem7.1, and is closelyrelated to Example8.1.

Example 8.6. Let X be a qcqs algebraic space, and assume{Ai} is a set of henselian local rings. SetA =

∏iAi. Then one can showX(A) ≃

∏iX(Ai) directly as follows. The argument for injectivity

in the proof of Theorem7.1 is elementary, and we offer no improvements here. For surjectivity, fix aNisnevich coverU → X with U an affine scheme. Given mapsai : Spec(Ai) → X, one may choose liftsai : Spec(Ai) → U asAi is henselian local. This shows

∏i U(Ai) →

∏iX(Ai) is surjective. AsU is

affine, one clearly hasU(A) =∏

i U(Ai), so the compositeU(A) → X(A) →∏

iX(Ai) is surjective, andhenceX(A) →

∏iX(Ai) is surjective.

We discuss one application of Theorem1.3 to describing adelic points on algebraic spaces over globalfields; in fact, only the significantly easier Example8.6 is used the proof, but we record the statement any-ways. First, we fix some notation (and adhere to standard conventions in number theory for any unexplainednotation). LetK be a global field,S a finite non-empty set of places ofK (assumed to contain the places at∞), AK the adele ring ofK, andAK,S ⊂ A the subring of adeles integral outsideS. Then we have:

Corollary 8.7. For any qcqs algebraic spaceX overOK,S, the natural map induces bijections

X(AK,S) ≃∏

v∈S

X(Kv)×∏

v/∈S

X(Ov).

If additionallyX is finitely presented overOK,S, then

X(AK) ≃∏

v∈S

X(Kv)×∏

v/∈S

′(X(Kv),X(Ov)

).

In the special caseX = Ga, Corollary8.7 is a definition. Slight variants of Corollary8.7 can also befound in work of Conrad [Con12a, page 613-615] and [Con12b, Theorem 3.6].

Proof. The first part is immediate from Example8.6asOv andKv are henselian local rings. For the second,note thatAK = colimAK,T , where the colimit runs over finite setsT of places containingS. As X isfinitely presented, one obtainsX(AK) = colimX(AK,T ). By definition of the restricted product, one alsohas

∏

v∈S

X(Kv)×∏

v/∈S

′(X(Kv),X(Ov)

)≃ colim

( ∏

v∈S

X(Kv)×∏

v∈T\S

X(Kv)×∏

v/∈T

X(Ov)),

where the colimit is indexed by the sameT ’s as before. The claim is now immediate from the first part.�

33

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