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arXiv:1304

.2520v1

[math.AG

]9Apr2013

A FUNCTORIAL FORMALISM FOR QUASI-COHERENT SHEAVESON A GEOMETRIC STACK

LEOVIGILDO ALONSO TARRO, ANA JEREMAS LPEZ, MARTA PREZ RODRGUEZ,AND MARA J. VALE GONSALVES

ABSTRACT. A geometric stack is a quasi-compact and semi-separated al-gebraic stack. We prove that the quasi-coherent sheaves on the small flattopology, Cartesian presheaves on the underlying category, and comodulesover a Hopf algebroid associated to a presentation of a geometric stackare equivalent categories. As a consequence, we show that the categoryof quasi-coherent sheaves on a geometric stack is a Grothendieck category.

We associate, in a 2-functorial way, to a 1-morphism of geometric stacksf : X Y, an adjunction f f for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes.This construction is described both geometrically in terms of the small flatsite and algebraically in terms of the Hopf algebroid.

CONTENTS

Introduction 11. Sheaves on ringed sites 42. Cartesian presheaves on affine schemes 8

3. Quasi-coherent sheaves on geometric stacks 114. Descent of quasi-coherent sheaves on geometric stacks 175. Representing quasi-coherent sheaves by comodules 226. Properties and functoriality of quasi-coherent sheaves 297. Describing functoriality via comodules 368. Deligne-Mumford stacks and functoriality for the tale topology 40References 45

INTRODUCTION

If one looks at the literature for the notion of quasi-coherent sheaf on analgebraic stack, one finds a somewhat confusing situation. For starters, the

Date: April 10, 2013.2000 Mathematics Subject Classification. 14A20 (primary); 14F05, 18F20 (secondary).This work has been partially supported by Spains MEC and E.U.s FEDER re-

search project MTM2008-03465 and MTM2011-26088, together with Xunta de GaliciasPGIDIT10PXIB207144PR .

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2 L. ALONSO, A. JEREMAS, M. PREZ, AND M. J. VALE

very notion of quasi-coherent sheaf does not refer to a ringed site but is of-ten mistook with a related notion, that of Cartesian presheaf. After [AJPV],

we became interested in the properties of its derived category. This is ourapproach to develop a sound basis for this study.As a starting point, the definition of quasi-coherent sheaf should be given

in the setting of a ringed site by an extension of the original definition givenin [EGA I, (5.1.3)]. After our research took off, we discovered that the StacksProject [SP] uses this kind of definition. The main technical problem is thelack of functoriality of the lisse-tale site. The Stacks Project solution is towork over the big flat sites. We do not follow this path because the categoryof sheaves of modules over a big (ringed) site does not possess a generator.This paper was developed independently of [SP], so we have tried to refermainly to [LMB] for the basics on stacks.

We assume always that the stacks we consider are semi-separated, i.e. the

diagonal is an affine morphism. On schemes this hypothesis is associated to anice cohomological behavior. The stacks that are furthermore quasi-compactare called geometric stacks in [L] and this will be our standing assumption.

A useful feature of geometric stacks is that they may be represented by apurely algebraic object, a Hopf algebroid. For such a stack X we consider acertain variant of the small flat site, that we will denote by Xfppf (see 3.5).Its associated topos is not functorial, but still, it has enough functorialityproperties to allow us to develop a useful formalism. Notice that it is finerthan the classical lisse-tale site, but has the advantage that a presentationyields a covering for every object in the small flat topology. This allows forsimpler proofs and the category of quasi-coherent sheaves does not change.

In the affine case there are three ways of understanding a quasi-coherent

sheaf, namely, as a sheaf of modules such that locally admits a presentation, as a cartesian presheaf1, and as a module over the ring of global sections.

These three aspects are, in some sense, present also in the global case. In thispaper we show that they arise quite naturally when we consider a geometricstack. For the small flat site, cartesian presheaves of modules are sheaves (infact, for any topology coarser than the fpqc) and moreover they are the sameas quasi-coherent sheaves. This is proved in Theorem 3.11. A geometric stackmay be described by a Hopf algebroid, as the stackification of its associatedaffine groupoid scheme. We prove that there is an equivalence of categories

between comodules over a Hopf algebroid and quasi-coherent sheaves on itsassociated geometric stack.This equivalence is developed in several steps. In section 4, we see that

quasi-coherent sheaves on a geometric stack correspond to descent data forquasi-coherent sheaves on an affine cover (Theorem 4.6). Incidentally, this is

1This property corresponds essentially to the fact that a localization of a module in anelement may be computed as a tensor product.

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QC-SHEAVES ON A GEOMETRIC STACK 3

the definition of quasi-coherent sheaf considered in the classic literature, e.g.[V1]. Once we are in the affine setting, we see that a descent datum corre-

sponds to certain supplementary structure on its module of global sections.By an algebraic procedure, in section 5, we show that this structure can betransformed into the structure of a comodule, establishing a further equiva-lence of categories. All these equivalences combined provide the equivalencebetween quasi-coherent sheaves on the geometric stack and comodules overthe Hopf algebroid (Corollary 5.9). An analogous result is due by Hovey un-der a different setting, as we will discuss later. An important consequence ofthis equivalence is that the category of quasi-coherent sheaves on a geometricstack is Grothendieck, i.e. abelian, with exact filtered direct limits and a setof generators.

The motivation for the results in the present paper is to have a convenientformalism for doing cohomology. So, it is crucial to have a good functoriality.

We devote section 6 to this issue. Specifically, we show that a map betweengeometric stacks induce a pair of adjoint functors between the correspondingcategories of quasi-coherent sheaves. This construction is not straightfor-ward because, as we remarked before, there is no underlying map of toposes,so the definition of the direct image has extra complications. Moreover, weshow that these constructions are 2-functorial. One needs to check in ad-dition that 2-cells induce a natural transformation between their associatedfunctors, a feature of stacks.

In section 7, we express the adjunction in a purely algebraic way. Thebenefit of this result is to put ourselves in the most convenient setting forapproaching cohomological questions. It is noteworthy that for a 1-morphismof geometric stacks f: X Y, the functor f has a nicer geometric description

while its adjoint f is easier to describe algebraically.In the final section we compare our functoriality formalism with the clas-

sical one. Let us recall that the category quasi-coherent sheaves on a schemedoes not change if we replace the Zariski by the tale topology. Moreover,the tale topology gives a reasonable topos for a geometric Deligne-Mumfordstack. So in this setting there is a couple of adjoint functors that restrict tothe classical ones in the scheme case and agree with the ones just defined inthe general case of a geometric stack. This is shown in Propositions 8.13, 8.14and Corollary 8.15.

In future work, we plan to apply the results obtained here to study thecohomology of geometric stacks.

Finally, let us compare our approach with previously published results.

The classical reference [LMB] defines quasi-coherent sheaf as a mixtureof Cartesian sheaf plus a local condition of quasi-coherence. Besides, it over-looks the lack of functoriality of the lisse-tale site. Olsson provides a remedy[O] but his pull-back functor has a complicated description through simpli-cial sites. Another approach is given by Hovey [Ho1], [Ho2]. His definition of

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quasi-coherent sheaf is in fact the notion of Cartesian presheaf, as he consid-ers mostly the discrete topology. So his theorem on the equivalence of quasi-

coherent sheaves and comodules over a Hopf algebroid is related to ours, butposed in a different framework. This point of view is also embraced by Goerssin [Go] who defines quasi-coherent sheaves trough the Cartesian condition.

We have to mention also Pribbles Ph. D. thesis [P]. We have used severaltimes the exact sequence associated to the monad (or triple) corresponding toa presentation. This has allowed us to simplify some arguments.

In [SP], they put very few restrictions on the diagonal, which makes thesituation more complicated. In the general setting a sheaf is quasi-coherentif, and only if, is Cartesian and locally quasi-coherent. As we have already re-marked, they work over big sites for which functoriality is easy but sacrificesother useful features for the usual constructions in cohomology.

Acknowledgements. We thank M. Olsson for the useful exchange concern-ing the algebraic structure of inverse images from which the consideration ofLemma 1.8 arose.

1. SHEAVES ON RINGED SITES

1.0. In this paper we will use as underlying axiomatics that of von Neu-mann, Gdel and Bernays using sets and classes. A category C has a classof objects and a class of arrows but the class of maps between two objetsA,B C, denoted HomC(A,B), is always assumed to be a set. For readersfond of Bourbaki-Grothendieck universes (and for comparison with results in[SGA 41]) fix two universes U V and identify elements ofU with sets and

elements ofV with classes.1.1. Ringed categories. Let C be a small2 category. We will denote byPre(C) :=Fun(Co,Ab) the category of contravariant functors from C to abeliangroups. We will usually refer to objects in this category as presheaves on C.We will say that (C,O) is a ringed category ifO is a ring object in Pre(C). Inother words O: Co Ring is a presheaf, or else O is a presheaf together witha couple of maps : OOO and 1: ZC O, with ZC denoting the constantpresheaf with value Z; this data makes commutative the usual diagramsexpressing the associativity and commutativity of the product and the factthat 1 is a unit.

1.2. Modules. Let (C,O) be a ringed category we will denote byPre(C,O)-Mod

or simply by Pre-O-Mod, if no confusion arises, the category of presheavesM Pre(C) equipped with an action ofO, OMM, that makes commu-tative the diagrams expressing the fact that the sections ofM are modulesover the sections ofO and the restriction maps are linear. We will say thatM is an O-Module.

2Its class of objects is a set, not a proper class.

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1.3. Cartesian presheaves. Let (C,O) be a ringed category. We will saythat an O-moduleM is a Cartesian presheaf if for every morphism f :X Y

in C theO

(X)-linear mapM(f)a :O(X) O(Y)M(Y) M(X)

adjoint to the O(Y)-linear map M(f) :M(Y) M(X) is an isomorphism. Wewill denote the category of Cartesian presheaves on (C,O) by Crt(C,O).

1.4. Ringed sites and topos. As usual we will call a small category C a siteif it is equipped with a Grothendieck topology [SGA 41, Exp. II 1]. If it isnecessary to make it explicit, we will denote the site by (C,), otherwise wewill keep the notation C. A site has associated a topos, its category of sheavesof sets. For a site (C,) we will denote its associated topos by C. We mayalso consider sheaves with additional algebraic structures, in particular, asheaf of rings O. We will call the pair (C,O) a ringed site and (C,O) a ringed

topos. We will always refer with the notation (C,O)-Mod or simply O-Mod tothe category ofsheaves of modules over the sheaf of rings O, and not to thebiggest category of presheaves of modules.

1.5. Let C be a site and X and object in C. We will consider the categorywhose objects are morphisms u : U X, and morphisms are commutativetriangles.3 We will denote such an object by (U, u), or simply by U if no con-fusion arises. The category C/X is a site. The coverings of the correspondingtopology of an object U are the families of morphisms {fi : Ui U} in C/Xthat are coverings in C. If (C,O) is a ringed site, then (C/X,O|X) is a ringedsite with the induced sheaf of rings O|X given by O|X(U, u) =O(U). IfM is asheaf ofO-Modules on the site (C,O) we define the sheaf restriction ofM to

C/X, denoted M|X, as the sheaf ofO|X-Modules on the site (C/X,O|X) givenby M|X(U, u) =M(U).

1.6. Quasi-coherent sheaves. Let (C,O) be a ringed site with a topology .We will say that a sheaf ofO-ModulesM is quasi-coherent if for every objectXC there is a covering {pi : Xi X}iL such that for the restriction ofM toeach C/Xi, there is a presentation

O|(I)Xi

O|(J)Xi

M|Xi 0,

where by F(I) we denote a coproduct of copies of a sheafF indexed by someset I. IfC is the site of open sets of a classical topology, this definition agreeswith [EGA I, 0, (5.1.3)]. We will denote the category of quasi-coherent sheaves

on (C,O) by Qco(C,O).1.7. Let f: (C,) (D,) be a continuous functor of sites [SGA 41, III, (1.1)]such that f preserves fibered products. In view of [SGA 41, III, (1.6)], thecontinuity of f is equivalent to the fact that it takes coverings in (C,) tocoverings in (D,).

3Sometimes called the comma category.

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Whenever C possesses colimits, this morphism induces a pair of adjointfunctors between the associated topos

Df1f

C.

Let us recall first the definition off. IfGD and VC, then

(fG)(V) =G(f(V)).

Let now FC, f1F is defined as the sheaf associated to the presheaf thatassigns to each UD the colimit

lim

IfU

F(V),

where IfU is the category whose objects are the pairs (V,g) with V C andg : U f(V) is a morphism in D, and whose morphisms

h : (V,g) (V,g)

are defined as morphisms

h : V V such that f(h)g =g

(see [SGA 41, I, 5.1] and [SGA 41, III, 1.2.1] where f1 is denoted fs). Noticethat, in general, f1 might not be exact. Iff1 is exact, we say that the pair(f, f1) is a morphism of topos from D to C according to [SGA 41, IV 3.1].

We want to extend this formalism to the case of ringed sites. To havethe possibility of transporting algebraic structures through the inverse imagefunctor, we need the following lemma, communicated to us by Olsson.

Lemma 1.8. Let I be a category with finite products and let F1, . . . ,Fr a finitefamily of functors Fi : Io Set, i {1. . . r}. The natural map

lim

I

(F1 Fr)

lim

I

F1

lim

I

Fr

is an isomorphism.

Proof. By using induction, we may assume that r = 2. Notice that

lim

I

F1 limI

F2

is identical tolimII

F1F2

where F1F2 : Io Io Set is the functor that sends a pair (U,V) Io Io toF1(U) F2(V). The diagonal functor : Io Io Io induces a map

z : lim

I

F1 F2 limII

F1F2 = limI

F1 limI

F2

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We will check first that z is surjective using the fact that I has products.Indeed, for any two objects U, V I and u F1(U), v F2(V) the element

[(u, v)] limII

F1F2

equals the class of (F1(p1)(u),F2(p2)(v)) F1(UV)F2(UV) via the naturalmaps p1 : U V U, p2 : U V V, so any element lies in the image of z.

Now, we prove that z is injective. Consider the functor : Io Io Io thatsends (U,V) to U V (notice that the product is taken in I). It gives thefollowing natural transformation

F1F2 (F1 F2)

defined as

F1(p1) F2(p2) : F1(U) F2(V) F1(U V) F2(U V)

on (U,V) Io Io. The functor induces a map

z : limII

F1F2 limI

F1 F2.

To see that z is injective is enough to check that zz = id. The map zz sendsthe class of (u1, u2) F1(U) F2(U) to the class of (F1(p1)(u1),F2(p2)(u2)) F1(U U) F2(U U). Let : U U U be the diagonal morphism. As weclaimed, the class of (F1(p1)(u1),F2(p2)(u2)) equals the class of the element

(F1(p1)(u1),F2(p2)(u2)) = (u1, u2).

1.9. Let f: (C,) (D,) be a continuous functor of sites as in 1.7. Let O be

a sheaf of rings on (C,) and P a sheaf of rings on (D,). Let us be given amorphism of sheaves of rings f# : O fP or, equivalently, its adjoint mapf# : f1OP.

Consider first the case in which (f, f1) is a morphism of topos. In this casewe say that ((f, f1), f#) is a morphism ofringed topos [SGA 41, IV, 13.1] andprovides an adjunction

P-Modf

f

O-Mod.

The functor f : O-ModP-Mod is defined by

fF :=Pf1O f1F

for a O-module F. Observe that for a P-module G, fG inherits a structureofO-module via f# and we have the claimed adjunction f f.

In general f1 may not be exact, and the previous discussion does not ap-ply. However, if we assume that the category IfU has finite products, then byLemma 1.8, the functor that defines the inverse image for presheaves com-mutes with finite products. The sheafification functor commutes also withfinite products since it is exact. It follows that f1 also commutes with finiteproducts. Thus, f1O has canonically the structure of a sheaf of rings and

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also f# is a homomorphism. By the same reason, ifF is an O-module thenf1F is an f1O-module. The functor f : O-ModP-Mod given by

fF :=Pf1O f1F

provides an adjunction f f as in the previous situation.Notice that we use here f1 for general sheaves and reserve f for modules

over ringed topos, contradicting the usage of [SGA 41, IV].

Remark. The perspicuous reader will notice that Lemma 1.8 is also implicitlyused in [O, Def. 3.11].

2. CAR TESI AN PRES HEAVE S ON AFFI NE SC HEMES

2.1. Let X = Spec(A) be an affine scheme. We will work with the category

Aff/X of flat finitely presented affine schemes over X, in other words theopposite category to the category of flat finite presentation A-algebras. Notethat this is an essentially small category. We will endow it with the topologywhose coverings are finite families {pi : Ui U}iL such that pi are flat finitepresentation maps such that iL Im(pi) = U. We will denote this site byAfffppf/X and by Xfppf its associated topos. Due to the fact that we limitthe possible objects to those flat of finite presentation, this site should notbe regarded as a big site and sometimes we will refer to it informally as thesmall flat site.

Let A or OX denote the ring valued sheaf defined by A(Spec(B)) =B. Notethat A is a sheaf on Afffppf/X as follows from faithfully flat descent [SGA 42,

VII, 2c], see also [SGA 1, VIII, 1.6] and [SGA 31, IV, 6.3.1]. We will denote

the corresponding ringed topos as (Xfppf, A) or (Xfppf,OX).Let Crt(X) :=Crt(Aff/X, A). We recall the somewhat classical

Proposition 2.2. With the previous notation, there is an equivalence of cate-gories

Crt(X) =A-Mod.

Proof. Consider the functors (X, ): Crt(X) A-Mod and ()crt : A-ModCrt(X) defined by (X,M) := M(X, idX) and Mcrt(Spec(R)) :=R A M. It isclear that they are mutually quasi-inverse.

Since each Cartesian presheaf in Crt(X) is isomorphic to a presheaf of the

form Mcrt, then it is a sheaf [SGA 42, VII, 2c]. The category Crt(X) is a fullsubcategory of the category of sheaves of A-Modules closed under the forma-tion of kernels, cokernels and coproducts, and it is an abelian category.

2.3. Let f: X Y be a morphism of affine schemes with X = Spec(B) andY= Spec(A). We have a continuous functor

fAff: Afffppf/YAfffppf/X

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given on objects by fAff(V, v) := (VY X,p2) with p2 denoting the projectionand on maps by the functoriality of pull-backs. It is clear that fAff preserves

fibered products, therefore by 1.7, induces a pair of adjoint functorsXfppf

f1f

Yfppf,

where we are taking the usual shortcuts f1 := (fAff)1 and f := (fAff).Moreover, we have the morphism of sheaves of rings

fAff# : A f Bgiven by fAff#(V, v) : R R AB, fAff#(V, v)(r) = r 1, for V= Spec(R). Since

the category IfAff

(U,u) has finite products, by 1.9 we have the adjunction

B-Mod

ff

A-Mod

where f := (fAff). Notice that if f is a flat finitely presented morphism,then fF(U, u) =F(U,f u).

Remark. We will abuse notation slightly and write G(U) for G(U, u) for any(U, u) Aff/X and GXfppf, if the morphism u is understood.

Proposition 2.4. It holds that

(i) IfFCrt(Y) then fFCrt(X).(ii) IfGCrt(X) then fGCrt(Y).

Proof. (i) Let (U, u) Afffppf/X with U= Spec(S). Denote IfAff

(U,u) by IU and let

AU := limIU

A(V).

The sheaf fF is the sheaf associated to the presheafP given by

P(U, u) = S AU limIU

F(V).

We have for every S-module Nthe following isomorphisms

HomS(S AU limIU

F(V),N) HomAU( limIU

F(V),N)

lim

IU

Hom A(V)(F(V),N)

lim

IU

HomS(S

A(V)F(V),N)

HomS( limIU

(S A(V)F(V)),N).

By Yonedas lemma, it follows that fF is isomorphic to the sheaf associatedto the presheafP such that

P (U, u) = lim

IU

S A(V)F(V).

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Let h : ((V, v),g) ((V, v),g) be a morphism in IU, with V = Spec(R), V =Spec(R) and h = Spec(). We have the commutative diagram

S R F(V) S RF(V)

S R R AF(Y) S R R AF(Y)

S AF(Y) S AF(Y)

id F(h)

idF(v)a idF(v)a

id id

where the top horizontal map is the transition morphism and the verticalmaps are isomorphisms by the definition of Cartesian presheaf and the prop-erties of the tensor product. Thus, P(U, u) = S AF(Y) and therefore fF isisomorphic to (B AF(Y))crt.

(ii) Let h : (V, v) (V, v) be a morphism in Afffppf/Y, V = Spec(R), V =Spec(R) and h = Spec(). We have to prove that the morphism

fG(h)a : R R fG(V

) fG(V)

is an isomorphism (see 1.3). We have that fG(V) =G(VY X) and similarlyfor fG(V). Notice that fG(h) =G(h Y idX), thus the morphism

G(h Y idX)a : (R AB) R ABG(V

Y X) G(VY X)

is an isomorphism becauseG is Cartesian. The result follows because fG(h)a

is identified with the composition of the following isomorphisms

R R G(V

Y X)

(R AB) R ABG(V

Y X)G(hid)a

G(VY X).Notice that f(G) is isomorphic to (AG(X))crt, where AG(X) means G(X) con-sidered as an A-module.

Corollary 2.5. A morphism of affine schemes f: X Y induces a pair ofadjoint functors

Crt(X)f

fCrt(Y).

Remark. We leave as an exercise for the interested reader to recognize thatthe previous adjunction agrees with the classical one

Qco(Xfppf,OX)f

fQco(Yfppf,OY),

via the fact that in both cases the (pre)sheaves can be represented by itsmodule of global sections. Notice that we consider the fppf topology insteadof the usual Zariski topology but this does not change essentially the categoryof quasi-coherent sheaves. We will see later (in Theorem 3.11) the agreementof quasi-coherent sheaves with Cartesian presheaves in a much more generalsetting that includes all quasi-compact and semi-separated schemes.

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3. QUASI-COHERENT SHEAVES ON GEOMETRIC STACKS

3.1. We will follow the conventions of [LMB], specially for the definition of

algebraic stack. But for our purposes we will restrict to what we call geomet-ric stacks after Lurie (see [L]). So, from now on, a geometric stack X will be astack on the tale topology of (affine) schemes over a base scheme (that willnot play any role in what follows)4 such that:

(i) the stack X is semi-separated, i.e. the diagonal morphism : X X X is representable by affine schemes;

(ii) the stack X is algebraic and quasi-compact, this amounts to the exis-tence of a smooth and surjective morphism p : X X from an affinescheme X.

We will refer to the map p : X X as a presentation ofX.A map representable by affine schemes is usually denominated an affine

morphism [LMB, 3.10]. Observe that every morphism u : U X

from anaffine scheme U is affine. Indeed, if v : V X is another morphism with Van affine scheme, we have that UX V X XX (U V) is 1-isomorphic to anaffine scheme and this 1-isomorphism is an isomorphism because UX V isin fact a category fibered in sets. We will identify it with the spectrum of itsglobal sections. We recall that the pair (X,XXX) with the obvious structuremorphisms is a scheme in groupoids whose associated stack is X, i.e.

X = [(X,XXX)]

with the notation as in [LMB, (3.4.3)].

Lemma 3.2. Let f: X Y a morphism of geometric stacks. The relative di-agonal X/Y : X X Y X is an affine morphism.

Proof. Consider the following composition

Xf

X Yp2

Y.

Notice that f p2 f. Semi-separatedness on schemes is preserved by basechange [AJPV, Proposition 2.2 (iii)], so by [LMB, Remarque (4.14.1)] it isalso preserved by base change on algebraic stacks, this guarantees the semi-separatedness of p2. The semi-separatedness off follows from the fact thatis a quasi-section, so its diagonal map is an equivalence.

Proposition 3.3. Let

X

Y

X Y

f

g g

f

be a 2-Cartesian square of algebraic stacks. If X, Y and Y are geometricstacks, then so is X.

4In fact, it can be shown that the fppf topology yields the same category of stacks.

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Proof. First, X is an algebraic stack by [LMB, (4.5)(i)].Let us check that the diagonal morphism X : X

X

X

is affine. Factor

it asX

X/X X

X X

h X

X

where h is the morphism in the following 2-Cartesian square

X X X

X

X

X X X

h

g g g

X

The morphism h is affine because it is obtained by base change from X. Themorphism X/X Y/Y X X

, so by Lemma 3.2, it is affine. It follows that X

is affine as wanted.Finally, given presentations p : X X and q : Y Y we have an inducedmorphism

p Y q : XY Y

X

Notice that XY Y is affine over X because is the base change of the mapq that is affine, therefore XY Y is an affine scheme. The morphism p Y qis smooth and surjective because both conditions are stable by base changeapplying [LMB, Prop. (4.13) et Rem. (4.14.1)] in view of [EGA I, (1.3.9)]. Thismakes of p Y q a presentation and the proof is complete.

3.4. For X a geometric stack, we define the category Aff/X as follows. Ob-jects are pairs (U, u) with U an affine scheme and u : U X a flat finitely

presented 1-morphism of stacks. Morphisms (f,) : (U, u) (V, v) are com-mutative diagrams of 1-morphisms of stacks. Let us spell this out. To givesuch an (f,) amounts to give a diagram

U V

X

f

u v

where f: U V is a morphism of affine schemes making the diagram 2-commutative, through the 2-cell : u v f. The composition of morphisms is

given by (g,) (f,) = (g f,f).

Notice that Aff/X is an essentially small category. In fact, let (U, u) Aff/Xwith U= Spec(B), by the semi-separateness ofX, the scheme UXX is affine.This affine scheme is isomorphic to the spectrum of a finitely presented alge-bra B over the ring of global sections of the structure sheaf of X. By faithfulflatness B is isomorphic to a subring ofB. This makes sense of what follows.

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3.5. The category Aff/X is ringed by the presheafO: Aff/X Ring, defined byO(U, u) =B with U= Spec(B). We define Cartesian presheaves on a geometric

stackX

as Cartesian presheaves over this ringed category,Crt(X) :=Crt(Aff/X,O).

Now, on Aff/X we may define a topology by declaring as covering the finitefamilies {(fi,i): (Ui , u i) (V, v)}iI, with every fi a flat finitely presentedmap, that are jointly surjective. It becomes a site that we denote Afffppf/X.The associated topos, i.e.the category of sheaves of sets onAfffppf/X is denotedsimply Xfppf. The siteAfffppf/X is a ringed site by the sheaf of rings associatedto the presheafO that we keep denoting the same.5 We define

Qco(X) :=Qco(Xfppf,O).

Remark.

(i) Every smooth morphism is flat of finite presentation, therefore oursite is finer than the usual lisse-tale topology, see [LMB, (12.1)].(ii) The topos Xfppf has enough points. The argument is the same as for

the lisse-tale topos, see [LMB, (12.2.2)].

In what follows we will abuse notation slightly and write F(U) forF(U, u)and analogously for any object in Aff/X if the morphism u : U X is obvious.

Lemma 3.6. A Cartesian presheafFCrt(X) is actually a sheafofO-Modu-les.

Proof. Let (V, v) Aff/X with V= Spec(C). Let {(fi,i) : (Ui, ui) (V, v)}iI bea covering with Ui = Spec(Bi). Let p

i j1 and p

i j2 be the two canonical projec-

tions from Ui VUj to Ui and Uj, respectively. Consider (Ui VUj, v fipi j1 )

Aff/X (notice that v fipi j1 = v fjp

i j2 ). We have the diagram

F(V)

iIF(Ui)

i,jIF(Ui VUj)2

1

l

where l = (F(fi,i))iI, for (xi)

iIF(Ui), we denote by 1(xi) the elementwhose component in F(Ui VUj) is F(p

i j1 ,

1i p

i j1 )(xi) and by 2(xi) the ele-

ment whose component inF(Ui VUj) isF(pi j2 ,

1j p

i j2 )(xj). Let us check that

this diagram is an equalizer. Being F Cartesian, this diagram is isomorphicto the diagram

F(V) iIBi CF(V) i,jI(Bi CBj) CF(V)2

1

l

where l(x) = (1 x)iI, for x F(V), and for (bi yi)iI

Bi CF(V), theelements 1((bi yi)iI) and

2((bi yi)iI) are those whose components in

(Bi CBj)CF(V) are bi 1yi , and 1 bj yj , respectively. Let B :=

iIBi

5The next lemma will show that there is no ambiguity.

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and the morphism : C B induced by the covering {fi : Ui V}. Since I isfinite, it suffices to prove that the diagram

F(V) B CF(V) (B CB) CF(V)j2 id

j1 id

l

is an equalizer, where l(x) = 1 x, j1(b) = b 1 and j2(b) = 1 b, for x F(V)and b B. As B is faithfully flat over C, the diagram

C

Bj2

j1

B CB

is an equalizer and stays so after tensoring by F(V) [SGA1, VIII 1.5]. Theresult follows.

Remark.

(i) The proof works for any topology coarser than fpqc, we have chosenthe fppf topology because it gives a reasonable small site.

(ii) The category Crt(X) is a priori a full subcategory ofPre-OX-Mod. Theprevious lemma implies that it is a full subcategory ofOX-Mod, asQco(X) is.

Proposition 3.7. The category Crt(X) is a full subcategory ofOX-Mod closedunder the formation of kernel, cokernels and coproducts, and therefore it is anabelian category.

Proof. Let K be the kernel of the morphism M M with M,M Crt(X)and (f,) : (U, u) (V, v) be a morphism in Aff/X with U= Spec(B) and V =Spec(C). If f is flat, we have the following commutative diagram

0 B CK(V) B CM(V) B CM(V)

0 K(U) M(U) M(U)

where the rows are exact and the midle and right vertical arrows are iso-morphisms because M and M are Cartesian sheaves. It follows that theremaining vertical arrow K(f,)a : B CK(V) K(U) is an isomorphism.

If f is not flat, let p : X X be an affine presentation with X = SpecA0.If p1 and p1, and p2 and p

2 are the projections from UX X and VX X to

U and V, and X, respectively, and : up1 p p2 and : vp1 p p2 are the

canonical 2-morphisms, we have a commutative diagram of affine schemes

UX X U

X VXX V

p1

p2 f f

p2 p1

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where the projections p1, p2, p1 and p2 are flat maps and

f =p1 . PutSpec(B) = UXX and Spec(C) = VXX. Since the projection p1 is faithfully

flat, to prove that the morphismK

(f,)

a

is an isomorphism it suffices toverify that the morphism

idB BK(f,)a : B B (B CK(V)) B

BK(U)

is an isomorphism. But this morphism is the composition of the followingchain of isomorphisms

B B (B CK(V)) B

C C

CK(V)

B C K(VX X,p p2) (via K(p

1, (

)1))

B C (C

A0 K(X)) (via K(p2,id))

B A0 K(X)

K(UXX,pp2) (via K(p2,id))B BK(U) (via K(p1,1))

The presheaf cokernel ofM M is a Cartesian presheaf and then it isthe cokernel in the category of sheaves ofO-modules. Similarly, the coproductof a set of Cartesian presheaves is a Cartesian presheaf and then it is thecoproduct in the category of sheaves ofO-modules.

Let X be a geometric stack and p : X X a presentation with X an affinescheme. Then p induces a continuous morphism of sites

pAff: Afffppf/X Afffppf/X

that is given on objects by pAff(U, u) := (UXX,p2), with p2 : UXXX theprojection, and on maps by the functoriality of pull-backs.

As discussed in 1.7 this morphism induces a pair of adjoint functors

Xfppfp1p

Xfppf.

Notice that p is a flat finite presentation 1-morphism, so (p1F)(V, v) =F(V,pv), for F Xfppf. In particular, p1OX = OX, therefore p1 inducesa functor between the categories of sheaves of O-Modules, i.e. p1 agreeswith the functor usually denoted p. So we have an adjunction as in 1.9

OX-Modp

pOX-Mod, (3.7.1)

with a particularly simple description of the functors involved, due to the factthat p induces a restriction functor between the corresponding sites.

Proposition 3.8. It holds that

(i) IfFCrt(X) then pFCrt(X).(ii) IfGCrt(X) then pGCrt(X).

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Proof. We prove (i). Let h : (U, u) (V, v) be a morphism in Afffppf/X withU= Spec(B) and V= Spec(C). We have to check that the morphism

(pF(h))a : B C pF(V, v) pF(U, u)

is an isomorphism. But pF(V, v) = F(V,pv) and analogously for (U, u).Therefore the previous morphism becomes

F(h, idpu)a : B CF(V,pv) F(U,pu)

where (h, idpu) : (U,pu) (V,pv) is a morphism in Afffppf/X. But this map isan isomorphism because F is Cartesian.

The proof of (ii) is similar to the proof given in Proposition 2.4 (ii).

Corollary 3.9. In this setting, the adjunction (3.7.1) restricts to

Crt(X)p

pCrt(X).

Let us now prove a converse to Proposition 3.8 (i).

Lemma 3.10. Let F be an OX-module. If pFCrt(X), then FCrt(X).

Proof. SetT :=pp. SinceF is a sheaf, we have the equalizer ofOX-modules

FF

TFTFTF

T2F

where is the unit of the adjunction p p. As a consequence of 3.8 TF,T2FCrt(X), it follows that FCrt(X).

Theorem 3.11. Let X be a geometric stack. The subcategories Crt(X) andQco(X) ofOX-Mod agree.

Proof. Let us prove first that Crt(X) Qco(X).Consider a presentation p : X X as before and F Crt(X). Let X =

Spec(A0). First, notice that the Cartesian sheafpF sits in an exact sequence

O(I)

X O(J)

X pF 0 (3.11.1)

that comes from a presentation ofM:=(X,pF) as A0-module applying thefunctor ()crt of Proposition 2.2. Let (U, u) Afffppf/X, and put V= UXX, p1and p2 being the projections and : up1 p p2 the canonical 2-morphism ofthe pullback. We have the covering {(p1,1) : (V,p p2) (U, u)} in Afffppf/Xwith p2 : VX a flat finite presentation map. Applying p2 to the resolution

(3.11.1) we obtain the exact sequence of sheaves ofOV-modules

O(I)V O

(J)V p

2p

F 0.

The result follows from the fact that the category OV-Mod is equivalent tothe category of sheaves ofO|(V,pp2)-modules over the site (Afffppf/X)/(V,pp2)by the comparison result [SGA 41, III, Thorme 4.1].

Let us prove now that Qco(X) Crt(X).

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SinceFQco(X), there exists a covering {(fi,i): (Ui , ui) (X,p)}iI suchthat for the restriction ofF to each (Afffppf/X)/(Ui, ui) there is a presentation

O|(I)(Ui ,ui)O|(J)(Ui ,ui )

F|(Ui ,ui ) 0

i.e. a presentation ofOUi -modules

O(I)Ui

O(J)Ui

uiF 0

uiF being the sheaf of OUi -modules given by uiF(W, w) = F(W, uiw) for

(W, w) Afffppf/Ui. The 2-morphism i : ui p fi induces an isomorphism be-tween the sheaves uiF and f

i p

F. Since OUi Crt(Ui), then u

iF Crt(Ui)

and thus fi pFCrt(Ui).

Because pF is a sheaf and {fi : Ui X}iI is a covering in Afffppf/X weobtain the equalizer ofOX-modules

pF iI fifi pF i,jI fi jfi jpF,

where fi j : Ui XUj X is the composition of the first projection with fi, or,what amounts to the same, the composition of the second projection with fj.Since fifi p

F,fi jfi jp

F Crt(X), then pF Crt(X). The result follows

from Lemma 3.10.

Remark. The consequence of the previous theorem is that not only all quasi-coherent sheaves are Cartesian (presheaves) but that this fact characterizesthem on a geometric stack. From now on, we will use freely this identifica-tion when dealing with quasi-coherent sheaves and use the most convenient

characterization for the issue at hand.

4. DESCENT OF QUASI-COHERENT SHEAVES ON GEOMETRIC STACKS

We proceed in our goal of describing quasi-coherent sheaves through al-gebraic data on an affine groupoid scheme whose stackification is the initialgeometric stack. As a first step, we will develop a descent result for quasi-coherent sheaves with respect to presentations of stacks.

4.1. The category of descent data. Let X be a geometric stack and p : XX a presentation with Xan affine scheme and G an OX-Module. As before, foran object (U, u) Afffppf/X, we will abbreviate G(U, u) byG(U) if no confusion

arises. Analogously, for a morphism (h,) we will shorten G(h,) by G(h).Consider the basic pull-back diagram

XXX X

X X

p2

p1 p

p

(4.1.1)

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A descent datum on G is an isomorphism t : p2G p1G ofOXXX-Modules

verifying the cocycle condition

p12 t p23 t =p13ton Afffppf/XX XX X, where as usual pi j : XX XX X XX X (i < j {1,2,3}) denotes the several projections obtained omitting the factor not de-picted. We define the category Desc(X/X) by taking as its objects the pairs(G, t) with G an OX-Module and t : p2G p

1G a descent datum on G; and

its morphisms h : (G, t) (G, t) with h : G G a morphism ofOX-Modulessuch that the diagram

p2G p1G

p

2

G p

1

G

t

p2 h p1 h

t

commutes.

4.2. Descent datum associated to a sheaf of modules. The 2-morphism : p p1 pp2 induces an isomorphism : p2p

p1p given by

(F

)(U, u) =F(idU,u), for FOX-Mod and (U, u) Afffppf/XXX.

Notice that for any OX-Module F, it holds that (pF,F) Desc(X/X). In-deed, consider the pullback

XXXXX X

XXX X

p3

p12 p

p p2

(4.2.1)

With the previous notations, we have that p13 = p12 and p23 = andthus, p23 p12 =p13. Now for (U, u) Afffppf/XXXX X we have

(p12F

p23F

)(U, u) =F(idU,p12u)F(idU,p23u)

=F(idU,p13u)

= (p13F

)(U, u).

This construction defines a functor

D : OX-ModDesc(X/X)

by D(F) := (p

F,F).

4.3. From descent data to sheaves of modules. We define a functor

G : Desc(X/X) OX-Mod

assigning to (G, t) Desc(X/X) the equalizer of the pair of morphisms

pGaG

b(G,t)

pp2p1G

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where

aG := (()p1G) (p1G)

b(G,t) := (pp2 t) (p2G)

with : pp1 pp2 denoting the functorial isomorphism and i the unitof the adjunction pi pi for i {1,2}. We denote this kernel by G(G, t). Thisconstruction is clearly functorial.

Proposition 4.4. The previously defined functors are adjoint, DG.

Proof. Let iG : G(G, t) pG be the canonical morphism. For each morphismofOX-modules g : FG(G, t), the adjoint to iGg : FpG through p pprovides the claimed morphism (pF,

F) (G, t) of descent data.

Reciprocally, if f: D(F) (G, t) is a morphism inDesc(X/X), the morphismF pG given by the adjunction p p factors through G(G, t) giving a

morphismFG(G, t), as desired.

To establish the next main result, Theorem 4.6, we will need a few techni-cal preparations. Let (V, v) Afffppf/X and consider the following pull-backs

Vp2 (XXX) XXX

V X

v2

v1 p2

v

VXX X

V X

w2

w1 p

pv

Denote W= Vp2 (XXX). Set h : VXXXXX the morphism such thatp1h = w2, p2h = vw1 and h =1. The morphism h : VXX W verifying

v1h

= w1 and v2h

= h is an isomorphism. Let us define : p2p

1 p

p

the isomorphism of functors given by (G)(V, v) =G(h) for each GOX-Mod.Set

c(G,t) := G (p2t) 2G.

Lemma 4.5. With the previous notations, the following identity holds

paG c(G,t) =pb(G,t) c(G,t).

Proof. Consider the pull-backs

W1 XX X

VXX X

2

1 p1

w2

W2 XXX

VX X X

2

1p2

w2

Let l : W2 VX X be the morphism such that w1l = w11, w2l = p12 and

l = 12 1. Put l

: W2 W1 for the morphism verifying that 1l = l

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and 2l = 2. Notice that l is an isomorphism and p(()p1G)(V, v) =G(l

).We have

(p(()p1G p1G) c(G,t)))(V, v) =G(l)G(1)G(h)t(W, v2)G(v1)=G(hl)t(W, v2)G(v1)

= t(W2, hl)G(hl)G(v1)

= t(W2, hl)G(w1l).

On the other hand, we have

(p(pp2 t p2G) c(G,t))(V, v) = t(W2,2)G(

1)G(h

)t(W, v2)G(v1)

= t(W2,2)t(W2, h

1)G(h

1)G(v1)

= t(W2,2)t(W2, h

1)G(w1l).

Thus, it is sufficient to prove that t(W2,2) t(W2, h1) = t(W2, hl ). For that,

consider the 2-pull-back

(XXX)pp2 X V V

XXX X

c2

c1 pv

pp 2

Let k : W2 (XX X)pp2 X V be the isomorphism given by c1k = 2, c2k =

w11, and k =11. Consider the morphism

id Xv : (XXX)pp2 X VXXXXX

satisfying p12(id Xv) = c1, p3(idXv) = vc2, and (idXv) = . The mor-phism g = (idXv)k is a finitely presented flat map. Applying the cocyclecondition oft to (W2,g), we obtain that

t(W2,p12g) t(W2,p23g) = t(W2,p13g)

and the result follows because p12g =2, p23g = h1 and p13g = hl .

Theorem 4.6. The functor D : OX-Mod Desc(X/X) (and, therefore its ad-joint G) is an equivalence of categories.

Proof. We will see first that , the unit of the adjunction DG, is an isomor-phism. Let FOX-Mod and consider the following commutative diagram

F ppF ppppF

G(pF,F

) ppF pp2p1pF

1

2

apF

b(pF,F

)

via 1F

F

ipF

where F is the unit of the adjunction p p applied to F. The first row isan equalizer because F is a sheaf and the claim follows.

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Let us prove now that , the counit of the adjunction D G, is an iso-morphism. Let (G, t) Desc(X/X). We have that (G,t) = G p iG, where G

denotes the counit of the adjunction p

p.

pG(G, t) ppG ppp2p1G

G

p aG

p b(G,t)

(G,t)

p iG

c(G,t)

Since p preserve finite limits, the horizontal row is exact and so by Lemma4.5 there exists a morphism r : G pG(G, t) with p iG r = c(G,t). It holdsthat c(G,t) (G,t) =p iG. Indeed,

c(G,t) (G,) = G p2t 2G G p iG

= ppG pp(G p2 t 2G) p

iG= ppG p

pG p(()p1G) p

p1G p iG.

and after an straightforward calculation we see that

ppG ppG p

(()p1G) pp1G = idppG .

We see now that r is the inverse of(G,t). Indeed,

p iG r (G,t) = c(G,t) (G,t) =p iG

and p iG is an monomorphism of sheaves, therefore r (G,t) = idG. Since G isa sheaf, 2G is injective and thus c(G,t) is an monomorphism of sheaves. From

c(G,t) (G,t) r =p iG r = c(G,t)

it follows that (G,t) r = idpG(G,t).

We denote by Descqc(X/X) the full subcategory of the category Desc(X/X)whose objects are the pairs (G, t) with GQco(X).

Corollary 4.7. The categories Qco(X) and Descqc(X/X) are equivalent.

Proof. The result follows because forFQco(X) we have D(F) Descqc

(X/X)and, conversely, for (G, t) Descqc(X/X) it holds that G(G, t) Qco(X) as fol-lows from Propositions 3.7, 3.8 and 4.6, having in mind Theorem 3.11.

Remark. Our method of proof is not too distant from Olssons [O, 4]. Notice,however, that he is working with Cartesian sheaves on a simplicial site whileour result follows from the more general Theorem 4.6 that applies not only toquasi-coherent Modules.

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5. REPRESENTING QUASI-COHERENT SHEAVES BY COMODULES

5.1. Hopf algebroids as affine models of geometric stacks. Let X be

a geometric stack and p : X X a presentation with X affine. The fiberedproduct XXX is an affine scheme because X is geometric. As we recalled in3, the pair (X,XXX) is a scheme in groupoids whose associated stack is X.

Let A0 and A1 be the rings defined by X = Spec(A0) and X1 := XX X =Spec(A1). Denote the pair A := (A0,A1). The dual structure of a schemein groupoids is called a Hopf algebroid. Let us spell out the correspondingstructure ofA, there are:

two homomorphisms

L,R : A0A1,

corresponding respectively to the projections p1,p2 : XX XX;

the counit homomorphism : A1 A0,

corresponding to the diagonal : XXXX; the conjugation homomorphism

: A1 A1,

corresponding to the map interchanging the factors in XXX; and the comultiplication

: A1 A1 R LA1,

corresponding, via the isomorphism

XX XXX

(XXX)p1 p2 (XXX)

induced by the projections p23 and p12, to the projection

p13 : XXXXXXXX,

expressing the composition of arrows in the scheme in groupoids(X,XX X).

These data is subject to the following identities:

(i) L = idA0 = R (source and target of identity).(ii) If j1, j2 : A1 A1 R LA1 are the maps given by j1(b) = b1, j2(b) =

1 b, then L = j1L and R = j2R (source and target of a com-position).

(iii) L = R and R = L (source and target of inverse).(iv) (idA1 ) = idA1 = ( idA1 ) (identity).(v) (idA1 ) = ( idA1) (associativity of composition).

(vi) If is the multiplication on A1, then ( idA1) = R and(idA1 ) = L (inverse).

(vii) = idA1 (inverting twice).

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Compare with [R, A1.1.1.] and [LMB, (2.4.3)].The geometric stack defined by a scheme in groupoids (Spec(A0),Spec(A1))

is denoted Stck(A) := [(Spec(A0),Spec(A1))]

(with the right hand side notation as in [LMB, (3.4.3)]). So, with the notationat the beginning of this section, X =Stck(A).

From now on, every geometric stack we consider will be the stack associ-ated to a Hopf algebroid. From a geometric point of view, the data of a Hopfalgebroid is more rigid than the underlying algebraic stack alone. It is equiv-alent to the datum of an algebraic stack together with a smooth presentationby an affine scheme. In any case, the algebroid determines the stack andwe will use this fact in what follows. Note that is an isomorphism (beinginverse to itself) and that L and R are smooth morphisms.

5.2. Comodules over a Hopf algebroid. Let A := (A0,A1) be a Hopf al-gebroid. A (left) A-comodule (M,M) is an A0-module M together with anA0-linear map M: M A1 R A0M, the target being a A0-module via L,and verifying the following identities:

(i) ( idM)M= (idA1 M)M (coassociativity of).(ii) ( idM)M = idM (M is counitary).

The map M is called the comodule structure map of M. If M and M are(left) A-comodules, a map of A-comodules is a A0-linear map : M M

such that the following square is commutative

M A1 R A0M

M A1 R A0M

M

id

M

We denote by A-coMod the category of (left) A-comodules. All comodulesconsidered in this paper will be on the left so we will speak simply of comod-ules.

If M is an A0-module one can always define a structure of A-comoduleon A1 R A0M by defining A1 R A0M := id. This structure is called theextended comodule structure on A1 R A0M.

Theorem 5.3. If A1 is flat as an A0-module, then the category A-coMod is a

Grothendieck category.Proof. In [R, A1.1.3.] it is proved that A-coMod is an abelian category if A1is a flat A0-module. Therefore what is left to prove is that this category hasexact direct limits and a set of generators.

Since tensor product commutes with colimits, the A0-module colimit of adiagram in A-coMod is an A-comodule. Direct limits of A-comodules areexact, because direct limits are exact in the category A0-Mod. Since A1 is flat

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as an A0-module, the kernel as an A0-module of a A-comodule morphisminherits the structure ofA-comodule.

Finally we show that the set S ofA-subcomodules of An

1 for n N

(viewedas A-comodules via ) is a set of generators ofA-coMod. Indeed, let Mbe aleft A-comodule, M= 0. The structure morphism = M: MA1 R A0Mis injective as a morphism of A-comodules, considering the structure of ex-tended comodule on A1 R A0M. Denote by q : A1 R A0MN the cokernelof and let p : A(I)0 Mbe a free presentation ofMas an A0-module. Since

the morphism id p : A1 R A0A(I)0 A1 R A0Mis a morphism of comodules

and A(I)1= A1 R A0A

(I)0 as comodules, we have a surjective morphism of co-

modules p : A(I)1 A1 R A0M. Given x M, x = 0, there are n N, y An1

such that p j(y) = (x) where j : An1 A(I)1 is the corresponding canonical

morphism. IfK is the kernel of q p j : An1 N, K S and y K, then ( p j)|K

factors through a non-zero map, i.e. HomA-coMod(K,M) = 0.

Remark. We remind the reader that if A is a Hopf algebroid arising from ageometric stack, A1 is smooth over A0 and, therefore, flat.

5.4. The category of descent data in A. Let A be a Hopf algebroid asbefore, and Man A0-module. Write A2 =A1 RL A1. A descent datum on Mis an isomorphism : A1 L A0MA1 R A0MofA1-modules veryfying thatthe diagram

A2 j1L A0 M A2 j1A1 A1 LA0 M A2 j1A1 A1 RA0 M

A2 A1 A1 LA0 M

A2 A1 A1 RA0 M A2 j2A1 A1 LA0 M

A2 R A0 M A2 j2A1 A1 RA0 M

id

j1R = j2L

id

id

(5.4.1)

commutes.We define the category Desc(A) by taking as its objects the pairs (M,)

with M an A0-module and a descent datum on M, and as its morphismsf: (M1,1) (M2,2) those homomorphism of A0-modules f: M1 M2 such

that the diagramA1 LA0 M1 A1 RA0 M1

A1 LA0 M2 A1 RA0 M2

1

id f id f

2

commutes.

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For future reference, note that commutativity of diagram (5.4.1) is equiva-lent to the following identity in A1 R L A1 R A0 M

ni=1

ac i b(1 xi) =n

i=1(a b)(ci) xi (5.4.2)

where a, b A1 and x M, with (1 x) =n

i=1 ci xi.

Proposition 5.5. The categories Desc(A) and A-coMod are isomorphic.

Proof. Let (M,) Desc(A). To give a structure of A-comodule on M onedefines = M: MA1 RA0 Mas the following composition

MLid

A1 LA0 M

A1 RA0 M.

Let us see that is counitary. Let x M, with notation as in 5.4.2 we have

( id)(x) =n

i=1(ci)xi.

To see thatn

i=1 (ci)xi = x, we apply the homomorphism id id to bothsides of the identity (5.4.2) for a = b = 1 to get

ni=1

(L (ci) xi) =n

i=1ci xi,

and since is an isomorphism we deduce thatn

i=1L(ci) xi = 1 x.

Thusn

i=1(ci)xi = ( id)

ni=1

L(ci) xi = ( id)(1 x) =x.

The coassociativity of follows from the identity (5.4.2) for a = b = 1. Now, iff: (M1,1) (M2,2) is a morphim in Desc(A), then

(idf)1 = (idf)1(L id) = 2(id f)(L id) = 2(L id)f =2f

so f: (M1,1) (M2,2) is a morphism of A-comodules. It follows that thisdefines a functor C : Desc(A) A-coMod.

Conversely, let (M,) be an A-comodule. We define

: A1 L A0MA1 R A0M

as the compositionA1 L A0M

id A1 L L A1 R A0 M

idA1 R A0M.

It easy to prove that is an homomorphism of A1-modules. We show that is an isomorphism by constructing its inverse. We define as the followingcomposition

A1R A0Mid

A1R LA1R A0Mid id

A1R R A1L A0Mid

A1L A0M.

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Let us see that and are mutually inverse. Consider the commutativediagram in which all tensor products are taken over A0 (with most subindicesomitted)

A1 R M A1 A1 M A1 A1 M A1 L M

A1 A1 M A1 A1 A1 M A1 A1 A1 M A1 A1 M

A1 A1 M A1 R M

id id id id

id id id id idid

id

id idid id id id

idid id

then we have

= ( id)(id id)(id id)(id id)(id)

= ( id)(id( id) id)(id )

= ( id)(idR id)(id) = id.

A similar computation shows that = id.Finally, it remains to prove that verifies (5.4.2). Since is coassociative,

we have for each x Mn

i=1ci (xi) =

ni=1

(ci) xi (5.5.1)

where (x) =n

i=1 ci xi. But (1 x) = (x), for each x M, and the resultfollows multiplying by a b A2 in both sides of (5.5.1).

Now, if f: (M1,1) (M2,2) is a morphism ofA-comodules, and 1 and2 are their corresponding descent data, then

(idf)1 = ( id)(id id f)(id1) = 2(idf)

and so f : (M1,1) (M2,2) is a morphism in Desc(A). It follows that thisdefine a functor D : A-coModDesc(A).

Note that the functors C and D are inverse to each other. In fact, let(M,) Desc(A) and : M A1R A0 M be its associated A-comodulestructure. The descent data associated to the comodule (M,) is the com-position ( id)(id ) = . Conversely, let (M,) be an A-comodule. If isthe corresponding descent data on M, the structure map of the A-comoduleassociated to (M,) is the morphism (L 1) =.

5.6. Let X = Spec(A) be an affine scheme. Recall that by Theorem 3.11 thecategoriesQco(X) and Crt(X) are the same. Correspondingly, the functors ()and ()crt are identified, too. We will stick to the former notation according tothe usage of [EGA I, (1.3)].

Proposition 5.7. The couple of functors

(X, ) : Qco(X) A0-Mod and () : A0-ModQco(X)

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induces an equivalence of categories between Descqc(X/X) and Desc(A), thatwe will keep denoting (X, ) and

().

Proof. If (G, t) Descqc(X/X), we will prove that the morphism

:= (G(p2)a)1 t1(X1, idX1 )G(p1)

a

is a descent datum on G(X).Consider the Cartesian square

X2 X1

X1 X

q2

q1 p2

p1

where X2 = Spec(A1 R LA1), q1 = Spec(j2) and q2 = Spec(j1). The multi-plication of the scheme in groupoids (X,X1) is the morphism m : X2 X1verifying p1m = p1q2, p2m =p2q1 and m = q1 q2, with notations as in(4.1.1). Let

w : X2 XXXXX (5.7.1)

be the isomorphism given by p12w = q2, p3w = p2 q1 and w = q1 with

as in 4.2.1. Its inverse w1, already defined on 5.1, is the unique morphismverifying q1w1 =p23 and q2w1 =p12. As a consequence

p12w = q2 p23w = q1 and p13w = m.

Thus, the cocycle condition of t on G reads

q

2t q

1t = m t

or, equivalently,t(X2, q2)t(X2, q1) = t(X2, m).

This last identity and the fact that t is a morphism of sheaves yields thecommutativity of (5.4.1).

Conversely, if (M,) Desc(A), using (5.4.1) it is easy to prove that thefollowing composition

p2M (A1 RA0 M) 1 (A1 LA0 M) p1Mis a descent datum on M. Theorem 5.8. The categories Descqc(X/X) and A-coMod are equivalent.

Proof. By Proposition 5.7 the category Descqc(X/X) is equivalent to Desc(A),which, in turn is equivalent to A-coMod by Proposition 5.5.

Remark. This result is similar to [Ho1, Theorem 2.2]. Notice however thatHoveys quasi-coherent sheaves over X are sheaves F for the (big) discretetopology while we work in the fppftopology. The ultimate reason of the agree-ment of both approaches lies in Lemma 3.6 that expresses the property of

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descent of Cartesian presheaves for this topology which makes them auto-matically sheaves. Notice also that the present proof is simpler once we have

made explicit in Proposition 5.7 that all the data involved refers to moduleson an affine scheme and, therefore, it has an essentially algebraic nature.

Corollary 5.9. For X = Stck(A), the categories Qco(X) and A-coMod areequivalent.

Proof. Combine Corollary 4.7 and Theorem 5.8.

We denote by Xp : Qco(X) A-coMod the equivalence of categories de-fined as Xp :=C(X, ) D, with D as in 4.2 and C as in Proposition 5.5. A

quasi-inverse is Xp =G () D.Corollary 5.10. For X = Stck(A), Qco(X) is a Grothendieck category.

Proof. Combine Theorem 5.3 with Corollary 5.9.

Proposition 5.11. We have

(i) If(M,) is an A-comodule and F is its associated OX-module, thenthe A0-module pF is M.

(ii) IfG Qco(X) then the comodule Xp (pG) is isomorphic to the ex-tended comodule A1 R A0 G(X).

Proof. By Theorem 5.8 (M,) corresponds to (M, t) Descqc(X/X). By Corol-lary 5.9 pF=pG(M, t) =M. This gives (i).

For (ii), we have by 2.2 that G(X) and G are isomorphic A0-modules andthen the comodules Xp (pG(X)) = A1 R A0G(X) and

Xp (pG) = G(X1,p2)

are isomorphic via G(p2)a : A1 R A0G(X) G(X1,p2). To prove (ii) is suf-ficient to show that the comodule structure on A1 R A0G(X) given by 4.2,Proposition 5.7 and Proposition 5.5, is id. Set M= G(X). The descentdatum on A1 R A0Mis defined by

= (pM(p2, id)a)1pM(id,1) (pM(p1, id))a.The structure map is the composition

A1 R A0MLid

A1 L LA1 R A0M

A1 R LA1 R A0M.

Consider the isomorphism w : X2 XXXXX in (5.7.1). We obtain

= (pM(p2, id)a)1pM(id,1)pM(p1, id)= (pM(p2, id)a)1pM(p1,1)= (pM(p2, id)a)1M(w)1M(w)M(p13)=M(m) = id.

the last equalities hold because M(w)pM(p2, id)a = id and m = Spec().

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6. PROPERTIES AND FUNCTORIALITY OF QUASI-COHERENT SHEAVES

We have reached our first main objective: representing quasi-coherent

sheaves on a geometric stack. It is time to move forward to the next one,namely, describing a nice functoriality formalism for these kind of a sheavesfor an arbitrary 1-morphism of stacks. Let then f: X Y be a 1-morphism ofgeometric stacks.

Unfortunately, f does not induce a (continuous) functor from Afffppf/Y toAfffppf/X, so we can not apply directly the methods from 1.7 and 1.9. Ofcourse, it could be done if f is an affine 1-morphism, but this condition isclearly too restrictive.

Besides, there is no hope to define a topos morphism from Xfppf to Yfppfbecause the topology is too fine, however we will construct an adjunction thatwill induce the corresponding pair of adjoint functors between the categoriesof quasi-coherent sheaves.

6.1. Direct image. For fixed (V, v) Afffppf/Y we denote by J(v,f) the cate-gory whose objects are the 2-commutative squares

U V

X Y

h

u v

f

(6.1.1)

where (U, u) Afffppf/X. We denote this object by (U, u, h,). A morphismfrom (U, u, h,) to (U, u, h,) is just a morphism (l,) : (U, u) (U, u) inAfffppf/X verifying hl = h and = l f. IfGPre(Afffppf/X) we define thepresheaf fG by:

(fG)(V, v) = limJ(v,f)

G(U, u).

It is a presheaf. Let (V1, v1), (V2, v2) Afffppf/Y and (j,) : (V1, v1) (V2, v2).In this situation we define a functor

(j,) : J(v1,f) J(v2,f)

by (j,)(U, u, h,) := (U, u, jh,h ). This induces the restriction morphism

(fG)(V2, v2) (fG)(V1, v1).

Notice that if f is an affine morphism, then fG(V, v) =G(X Y V,p1) be-cause (XY,p1,p2,0), with 0 the canonical 2-cell in the 2-pullback, is anfinal object ofJ(v,f).

6.2. Inverse image. Let (U, u) Afffppf/X, we denote by I(u,f) the categorywhose objects are the 2-commutative squares as 6.1.1 with (V, v) Afffppf/Y.We denote these objects now as (V, v, h,). A morphism from (V, v, h,) to

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(V, v, h,) is a morphism (j,) : (V, v) (V, v) inAfffppf/Y verifying jh = h

and =h . For FPre(Afffppf/Y), the presheaf fpF is given by

(fpF)(U, u) = limI(u,f)

F(V, v)

for each (U, u) Afffppf/X. It is also a presheaf. Consider (U1, u1), (U2, u2) Afffppf/X together with (l,): (U1, u1) (U2, u2). In this situation we definea functor

(l,) : I(u2,f) I(u1,f)

by (l,)(V, v, h,) := (V, v, hl ,l f). This induces the restriction morphism

(fpF)(U2, u2) (fpF)(U1, u1).

If f is a flat finitely presented morphism, then, fpF(U, u) =F(U,f u) be-cause (U,f u, idU, id) is an initial object ofI(u,f).

Lemma 6.3. Let f1,f2 :X

Y

be 1-morphisms of geometric stacks. In theprevious setting, any 2-morphism : f1 f2 induces equivalences of categories

(i) J : J(v,f1) J(v,f2),(ii) I : I(u,f2) I(u,f1).

Proof. For (i), define, using the notation as in 6.1,

J(U, u, h,) = (U, u, h,1u)

and extend in an obvious way to maps. We see that these data defines afunctor and a quasi-inverse is given by (1)J.

Analogously, for (ii), define, using the notation as in 6.2,

I(V, v, h,) = (V, v, h,u)

As before, this defines a functor whose quasi-inverse is (

1)I. Proposition 6.4. Let f1,f2 : X Y be 1-morphisms of geometric stacks. Withthe previous notation, given a 2-morphism : f1 f2 we have the followingisomorphisms

(i) : f1 f2,(ii) p : fp2 f

p1 .

Proof. To prove (i), apply the equivalence J : J(v,f1) J(v,f2) to the setsG(U, u) with (U, u) J(v,f1), this gives a map

lim

J(v,f1)G(U, u) lim

J(v,f2)

G(U, u).

This isomorphism is natural, therefore induces a natural isomorphismf1G f2G.

For (ii) use a similar argument with I in place ofJ.

Proposition 6.5. The previous constructions define a pair of adjoint functors

Pre(Afffppf/X)fp

fPre(Afffppf/Y).

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Proof. We have to prove, for F Pre(Afffppf/Y) and G Pre(Afffppf/X) thatthere is an isomorphism

HomX(fpF,G) HomY(F,fG)

A map fpF G induces for every (U, u) Afffppf/X and (V, v) I(u,f) mor-phisms

F(V, v) lim

I(u,f)F(V, v) G(U, u)

Now fixing (V, v) and varying (U, u) we get a map

F(V, v) lim

J(v,f)G(U, u).

This construction provides the adjunction map. By a similar procedure, we

obtain a map in the opposite direction. It is not difficult to check that bothmaps are mutually inverse.

6.6. The continuity of our construction is expressed by the relation betweenthe coverings. Let us spell it out.

Take a covering {(gi,i) : (Vi, vi) (V, v)}iI in the site Afffppf/Y. Given(U, u, h,) J(v,f), consider for each i I the following Cartesian square ofaffine schemes

UV Vi Vi

U V

hi

gi gi

h

Set Ui := UVVi and i : f ugi vi h i, given by i :=1i h i g

i . This defines

a functor

gi : J(v,f) J(vi,f) (6.6.1)

by gi(U, u, h,) := (Ui, ugi, hi,i).A consequence of this observation is the following important statement.

Proposition 6.7. IfG is a sheaf on Afffppf/X, then fG is a sheaf.

Proof. Let {(gi ,i): (Vi , vi) (V, v)}iI be a covering in the site Afffppf/Y.With the previous notation, given (U, u, h,) J(v,f), the family

{(gi, id) : (Ui , ugi) (U, u)}iI

is a covering in the site Afffppf/X. Associated to the sheafG we have anequalizer diagram

G(U)

iIG(Ui)

i,jIG(Ui UUj).2

1

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Taking limits, we obtain another equalizer diagram (in the following we willkeep the notation on the maps to avoid the clutter)

lim

J(v,f)G(U) lim

J(v,f)

iIG(Ui) lim

J(v,f)

i,jIG(Ui UUj).

2

1

That, by commutation of limits [Bo, Prop. 2.12.1], it may be rewritten as

lim

J(v,f)G(U)

iI lim

J(v,f)G(Ui)

i,jI lim

J(v,f)G(Ui UUj).

2

1

Let us denote by Ji(v,f) the essential image of J(v,f) in J(vi,f) through thefunctor gi (6.6.1). Note that

limJ(v,f)

G(Ui) = limJi(v,f)

G(Ui)

All of this applies verbatim to Ji j(v,f).Consider now the following commutative diagram

lim

J(v,f)G(U)

iI lim

Ji(v,f)G(Ui)

i,jI lim

Ji j (v,f)G(Ui UUj)

fG(V)

iI fG(Vi)

i,jI fG(Vi V Vj)

2

1

2

1

can can

In a natural way, for every U= (U, u, h,) J(vi,f)wehave(U, u,g1h,h) J(v,f). Then we get

Ui =gi(U, u,g1h,h ) Ji(v,f)

jointly with maps in J(vi,f), U Ui and Ui U whose composition is theidentity. This implies that the canonical map

fG(Vi)can

lim

Ji(v,f)G(Ui)

is a monomorphism. Similarly the rightmost vertical map is an monomor-phism, too. The top row is an equalizer and this implies that the bottom rowis, as a consequence, fG a sheaf.

Corollary 6.8. Let F Yfppf. We denote by f1F the sheaf associated to thepresheaf fpF. There is a pair of adjoint functors

Xfppf

f1f

Yfppf.

Proof. Combine the previous Propositions 6.5 and 6.7, having in mind theadjoint property of sheafification.

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Remark. Notice, however, that, in general, the pair (f1,f) does not define amorphism of topoi, since f1 need not be left exact. For an explicit example

see [Be, 4.4.2]. In the case in which f is a flat finitely presented morphism,then f1 is exact.

Proposition 6.9. Let f: X Y and g : Y Z be 1-morphisms of geometricstacks. It holds that (g f) =gf

Proof. Let F Xfppf. For each (W, w) Afffppf/Z, we have that

(g f)F(W, w) = limJ(w,g f)

F(U, u)

where J(w,g f) is as in 6.1 for (W, w). On the other hand

gfF(W, w) = limJ(w,g)

fF(V, v)

We have a family of maps compatible with morphisms in J(w,g)(g f)F(W, w) fF(V, v)

defining a natural map

(g f)F(W, w) gfF(W, w). (6.9.1)

Let us show that this map is an isomorphism.Next, for each (U, u, h,) J(w,g f) we will construct a map

gfF(W, w) F(U, u)

compatible with morphisms in J(w,g f). Consider the 2-fibered product stackY

:= Y Z W, its canonical projections q1 and q2, and 0 : gq 1 wq2 itsassociated 2-morphism. Y is a geometric stack by Proposition 3.3. There is

a morphism f : U Y given by the 2-isomorphism : g f u wh, such thatq1f = f u, h = q2f and 0f = . Let q : V Y

be a presentation. Considerthe 2-fiber squares

U V

U Y

f

q q

f

V V

V Y

p2

p1 q

q

Denote by v : V Y the map v = q1q. It is clear that (V, v, q2q,0q) J(w,g). Take v := vp1. In J(w,g), we have the projections (p

1, id) and

(p

2, q1

) from (V

, v

, q2qp

1,0qp

1) to (V

, v

, q2q,0q). We have a diagram

gfF(W, w) fF(V, v) fF(V, v)

F(U, u) F(U, u) F(U UU, u)

fF(p1 , id)

fF(p2, q1)

F(p1, id)

F(p2, id)

F(q, id)

can can

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where p1,p2 : U UU U are the canonical projections, u = uq, u = up1and the vertical maps labelled can are induced by the limit diagrams in-

dexed by J(v

,f) and J(v

,f). The map is the canonical map correspondingto the limit of the diagram indexed by J(w,g).Notice that q is smooth and surjective and the same holds for q therefore,

{q : U U} is a (flat) covering in Afffppf/X. As F is a sheaf, the lower row isan equalizer diagram. We have that fF(p1, id)= fF(p

2, q1

), thereforethere is a canonical vertical dashed map. Passing to the limit along J(w,g f)we obtain a natural map

gfF(W, w) (g f)F(W, w)

which is inverse to 6.9.1.

Corollary 6.10. In the previous situation (g f)1 f1g1.

Proof. It is is a consequence of the previous proposition by adjunction.

6.11. In general, for f: X Y, there is no map between ringed sites, however,we consider the morphism of sheaves of rings f# : OY fOX, given for each(V, v) Afffppf/Y by the induced ring homomorphism

f#(V,v) : OY(V, v) limJ(v,f)

OX(U, u).

We want to extend the previous adjunction to the categories of sheaves ofmodules. As we do not have neither a topos morphism neither a continuousmap of sites, this extension is not straightforward.

The functor f takes OX-Modules to fOX-Modules and, by the forgetfulfunctor associated to f#, to OY-Modules. Note that the formula in Proposition

6.9 keeps holding by the transitivity of the forgetful functors.We will apply Lemma 1.8 to guarantee that the functor fp commutes withfinite products and so does f1. It will follow that ifF is an OY-module,f1F will be an f1OY-module. Therefore we have to check the hypothesisof lemma 1.8, so we are reduced to prove the following.

Lemma 6.12. The category I(u,f) has finite products.

Proof. We only need to prove that any two objects in I(u,f) admit a product.For that, let (V1, v1, h1,1) and (V2, v2, h2,2) be objects in I(u,f). For i {1,2}, let pi be the projection from V1 Y V2 to Vi, and : v1p1 v2p2 bethe canonical 2-morphism. If h : U V1 Y V2 is the morphism verifyingp1 h = h1, p2 h = h2 and h = 2 11 , then the object (V1 Y V2, v1p1, h,1)

together with the morphisms (p1, id) and (p2,) is a product of (V1, v1, h1,1)and (V2, v2, h2,2) in I(u,f).

Corollary 6.13. The functor f : OY-ModOX-Mod given by

fF :=OX f1OY f1F

is left adjoint to the functor f : OX-ModOY-Mod.

Proof. Combine Corollary 6.8 with the previous discussion.

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Corollary 6.14. Let f: X Y and let g : Y Z be 1-morphisms of geometricstacks. It holds that (g f) = fg.

Proof. It is a consequence of Proposition 6.9 by adjunction.

Corollary 6.15. Let f1,f2 : X Y 1-morphisms of geometric stacks. With theprevious notation, given a 2-morphism : f1 f2 we have isomorphisms

(i) : f1 f2(ii) : f2 f

1

as functors defined on OX-Mod and OY-Mod, respectively.

Proof. Immediate from 6.4 and the previous discussion.

Remark. Notice that the isomorphism in 4.2 is a particular case of (ii).

Proposition 6.16. IfFQco(Y) then fFQco(X).

Proof. Let p : XX

and q : YY

be affine presentations such that we havea 2-commutative square

X Y

X Y

f0

p q

f

It follows from the previous Corollary that pfF = f0 qF, via using

pseudofunctoriality. By Proposition 3.8 qFCrt(Y) and by Proposition 2.4,f0 q

FCrt(X). We conclude applying Lemma 3.10 and Theorem 3.11.

Proposition 6.17. IfGQco(

X

) then fGQco(

Y

).Proof. If f is an affine morphism the proof is similar to the given in Propo-sition 2.4. If f is not affine, take an affine presentation p : X X. Themorphism f p is affine. Let T := pp as in Lemma 3.10 and consider thefollowing equalizer of quasi-coherent sheaves

GG

TGTGTGT2G.

Since f is left exact we have the equalizer ofOY-modules

fGfG

fTGfTG

fTG

fT2G.

From Proposition 3.8 and Theorem 3.11 by using Proposition 6.9 it followsthat fTG and fT2G are quasi-coherent sheaves. Then, fG is a quasi-coherent Module since it is the kernel of a morphism of quasi-coherent Mod-ules.

Corollary 6.18. There is a pair of adjoint functors

Qco(X)f

fQco(Y).

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Proof. Combine the adjunction of Corollary 6.13 with Propositions 6.16 and6.17.

7. DESCRIBING FUNCTORIALITY VIA COMODULES

Let : A B be a homomorphism of Hopf algebroids. Following [Ho2],we describe a pair of adjoint functors

B-coMod A-coMod.B0 A0

U

We will see that for the corresponding 1-morphism of stacks f := Stck()

f: Stck(B) Stck(A),

this adjunction corresponds to the previously considered f f betweensheaves of quasi-coherent modules on the small flat site.

7.1. A homomorphism of Hopf algebroids : A B is a pair of ring homo-morphisms i : A i Bi with i {0,1} that respects the structure of a Hopfalgebroid, namely, the following equalities hold

1L = L0 1 =0

1R = R0 1 =1 (1 1) = 1.

7.2. The homomorphism : A B induces a functor

B0 A0 : A-coModB-coMod,

sending an A-comodule (M,) to the B-comodule (B0 A0 M,) where is

given by the composition

B0 A0 Mid

B0 L A1R A0 Mid

B1 R A0MB1 R B0 (B0 A0 M)where (b0 a1) := L(b0)1(a1), for a1 A1 and b0 B0.

7.3. From [Ho2, Proposition 1.2.3], we will describe

U : B-coModA-coMod,

a right adjoint for the functor B0 A0 . This will be done in two steps.Consider first an extended comodule, that is, given N B0-Mod, let us

consider the comodule B1 B0 Nwith the induced structure. We define

U(B1 B0 N) :=A1 A0 N

with the induced structure. For maps : B1 B0 NB1 B0 N with N,N

B0-Mod, then U() is defined as the following composition

A1 R A0Nid

A1 R LA1 R A0N

id1idA1 R L0B1 R B0 N

idA1 R L0B1 R B0 N

ididA1 R A0N

.

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Now, let (N,) be an arbitrary B-comodule. Let : B1 B0 NN be the

cokernel of as B0-modules. Let be the induced comodule structure on N

and consider the following A-comodule homomorphism:A1 R A0N

U()A1 R A0N

.

Define

U(N) := ker(U())

The extension to morphisms is straightforward by the naturally of kernels.

Proposition 7.4. Given a homomorphism : A B of Hopf algebroids,there is an associated pair of adjoint functors

B-coMod A-coMod.B0 A0

U

Proof. Let (P,) B-coMod and (M,) A-coMod. We have to constructan isomorphism

HomB-coMod(B0 A0 M,P) HomA-coMod(M,U(P))

First, ifP is extended, i.e. P =B1 B0 N for an B0-module N, then we havethat

HomB-coMod(B0 A0 M,P) = HomB-coMod(B0 A0 M,B1 B0 N)= HomB0-Mod(B0 A0 M,N)= HomA0-Mod(M,N)= HomA-coMod(M,A1 A0 N)

= HomA-coMod(M,U(P))

In the general case, P may be obtained as a kernel of a morphism betweenextended B-comodules, in other words, we have an exact sequence

0 P P P

with P and P extended B-comodules. Now we have a diagram

HomB (B0 A0 M,P) HomB (B0 A0 M,P) HomB (B0 A0 M,P

)

HomA

(M,U

(P)) HomA

(M,U

(P )) HomA

(M,U

(P ))

in which the last two vertical maps are isomorphisms because P and P areextended comodules, therefore they induce the dashed vertical map which isalso an isomorphism as desired.

Remark. The previous result is due to Hovey [Ho2]. We have included a prooffor the readers convenience.

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7.5. Our next task is to identify the functor U with the direct image ofsheaves on the geometric stacks corresponding to the Hopf algebroids.

Let f:X

Y

be a morphism of geometric stacks, p : XX

and q : Y Y

affine presentations and f0 : X Y be a morphism such that the followingsquare

X Y

X Y

f0

p q

f

is 2-commutative. Set X = Stck(B) and Y = Stck(A). Denote the ring homo-morphisms associated to f as 0 : A0 B0 and 1 : A1 B1. We remind thereader that = (0,1) is a homomorphism of Hopf algebroids (see 7.1).

Proposition 7.6. There is a natural isomorphism of functors from Qco(X) toA-coMod

: Yq f U

Xp

Proof. We will start considering an object in the source that lies in the es-sential image of the functor T = pp. Let F Qco(X) with pF = M, forMB0-Mod. By Proposition 5.11(ii), XpTF= (B1 B0 M, id). By 7.3

UXpTF= (A1 A0 M, id)

On the other hand, by 2-functoriality of direct images there is a canonicalisomorphism via 1

fTF= fppF qf0M= qM[0].

Therefore there is an isomorphism Yq fTFYq qM[0] and we have

Yq qM[0] = (A1 A0 M, id),

by Proposition 5.11(ii), again. This defines TF : Yq fTF U

XpTF. No-

tice that T() is a natural transformation.

Finally we extend this to all quasi-coherent sheaves onX

and all mor-phisms. Consider the canonical equalizer of quasi-coherent OX-modules

FF

TFTFTF

T2F.

We may thus present every object in Qco(X) as a kernel of objects that liein the image of the functor T. Consider the following commutative equalizer

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diagram

Y

q fF Y

q fTF Y

q fT2F

UXpF U

XpTF U

XpT

2F.

F

Yq fTF

Yq fTF

UXp TF

UXpTF

TF T2F

Yq fF

UXp F

Notice that UXpTFTF =T2FYq fTF being T() a natural transfor-

mation. Let us check that

UXpTFTF =T2FYq fTF.

Applying Xp to the map TF : TFT2F, we get

F(p13, id) : F(X1,pp2) F(X1 XX,pp3).

In other words, having in mind that p13w = m where w : X2 X1 XX is theisomorphism in 5.7.1 and m = Spec(), taking M:= F(X,p), this morphismcorresponds to:

id : B1 R A0MB1R L B1R A0 M.

Now, we apply U and we obtain the following composition

A1 R A0Mid

A1 R LA1 R A0Mid1id

A1 R LB1R B0 M.

On the other hand, consider the 2-cartesian squares

YY X X

Y Y

2

1 f p

q

1

(YY X) X X X

YY X X

2

1p

p2

2

and the morphism l : (YY X) X X YY X. satisfying that 1l = 11,2l = 2 and 1l = f2 1

1. Applying

Yq f to TF, we have the following

commutative diagram

Yq fTF

Yq fT

2F

F(YY X,p2) F((YY X) X X,p2)

Yq fTF

via via

F(l, id)

Consider now the diagram

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(YY X) XX YY X

Y1 p2 f0p1X1 Y1 p2 YX

l

c c

l

where c is the composition of the following isomorphisms

Y1 p2 YX= (YY Y)p2 YX Yqq f0X YY X,

c is the isomorphism given by

Y1 p2 f0p1X1 (Y1 p2 f0X) X Xcid

(YY X) XX

and l = c1l c. A computation shows that the following composition

Y1 p2 f0p1X1 Y1 p2 f0p1X1 p2 XX

idf1id Y1 p2 p1 Y1 p2 YX

sYX (Y1 p1 p2 Y1)YX

mid Y1 p2 YX

agrees with l, where s is the isomorphism that interchanges the factorsof the fiber product. But then, considering the isomorphisms Y1 Y X =Spec(A1 R A0B0) and Y1 Y X1 = Spec(A1 R LB1), l

is identified the spec-trum of the composition

A1 R A0B0id

A1 R LA1 R A0B0id 1id

A1 R LB1R B0 B0.

Finally, as the map F(l, id) corresponds to the last map tensored with M=F(X,p) this shows the agreement of both morphisms.

The fact that the rows are equalizers defines the promised dashed naturalisomorphismF.

Corollary 7.7. There is a natural isomorphism of functors

Xpf

B0 A0

Yq .

Proof. Denote by Yq a quasi-inverse ofYq . The functor

Xpf

Yq is left ad-

joint to U, so it has to agree with B0 A0 .

8. DELIGNE-MUMFORD STACKS AND FUNCTORIALITY FOR THE TALE

TOPOLOGY

In section 6 we have shown how to obtain an adjoint functoriality of thecategory of quasi-coherent sheaves on a geometric stack overcoming the dif-ficulty of the lack of functoriality of the fppf topos. But one can (and should)wonder how this construction is related to the case where a functorial toposexists. This is the case for the tale topos on a Deligne-Mumford stack, asfollows from Theorem 8.7 below. We recall that the corresponding theory of

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quasi-coherent sheaves is equivalent to the one in the Zariski topology when-ever the stacks are equivalent to schemes [SGA 42, VII, 4.3]. The answer is

that our construction agrees with it. Let us see how.8.1. Let X be a geometric Deligne-Mumford stack and consider the categoryAffet/X whose objects are pairs (U, u) with U an affine scheme and u : U Xan tale morphism, morphisms are as in Aff/X. Note that all morphisms aretale. We define a topology in this category having as coverings finite familiesof tale morphisms that are jointly surjective. We denote also by Affet/Xthis site and by Xet the associated topos of sheaves. The category Affet/X isringed by the sheaf of rings O: Affet/X Ring, defined by O(U, u) =B withU= Spec(B). We define

Qcoet(X) :=Qco(Affet/X,O).

As in Theorem 3.11, Qcoet(X) agrees with the category of Cartesian sheaves,

and we will use this fact freely.

Let f: X Y be a 1-morphism of geometric Deligne-Mumford stacks.

8.2. Direct image for the tale topology. Let (V, v) Affet/Y. Denote byJet(v,f) the category whose objects are the 2-commutative squares

U V

X Y

h

u v

f

(8.2.1)

where (U, u) Affet/X. As in 6.1 (with appropriate changes) we define forGPre(Affet/X),

(fet G)(V, v) = limJet(v,f)

G(U, u).

Again, if f is an affine morphism, then (fet G)(V, v) =G(VY X,p2).

8.3. Inverse image for the tale topology. Now, fix (U, u) Affet/Y anddenote by Iet(u,f) the category whose objects are the 2-commutative squaresas in diagram (8.2.1), with (V, v) Affet/Y. By a similar procedure to 6.2define for FPre(Affet/Y) and (U, u) Affet/X

(fpF)(U, u) = lim

Iet(u,f)F(V, v).

Notice that if f is tale, then (fpF)(U, u) =F(U,f u).

Proposition 8.4. The previous constructions define a pair of adjoint functors

Pre(Affet/X)fpet

fet

Pre(Affet/Y).

Proof. The proof is completely similar to the one in Proposition 6.5.

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Proposition 8.5. The functor fp :Pre(Affet/Y) Pre(Affet/X) is exact.

Proof. The functor fp is right exact, because it has a right adjoint. To prove

that fp is left exact it is enough to show that Iet(u,f) is a cofiltered categoryfor (U, u) Affet/Y.

By [SGA 41, I, 2.7] we have to prove that the category Iet(u,f) is connectedand satisfies PS 1) and PS 2) in loc. cit. The category Iet(u,f) is connectedsince it has finite products (analogous to the fppfcase, Lemma 6.12).

To prove PS 1), for i {1,2} take (j i,i) : (Vi, vi, h i,i) (V, v, h,) mor-phisms in Iet(u,f), the projections pi : V1 V V2 Vi and h : U V1 V V2the morphism verifying pi h = hi. We have that (V1 V V2, v1p1, h,1) is inIet(u,f) and the morphisms (p1, id) and (p2,12 p2 1p1) satisfy

(j1,1) (p1, id) = (j2,2) (p2,12 p2 1p1).

Finally, we prove PS 2): Let (g,), (g

,

) : (V, v, h,) (V

, v

, h

,

) be mor-phisms in Iet(u,f) and consider the pull-back

V Y V V

V Y

p2

p1 v

v

We have the morphism r : V V Y V verifying p1r = g, p2r = g and r = 1. Consider the Cartesian square

Z V

V V Y V

q2

q1

r

and the morphism l : U Z such that q1l = h and q2l = h. Notice thatq1 is tale because is an open embedding and therefore, tale. The object(Z, vq1, l,) Iet(u,f) and the morphism (q1,id): (Z, vq1, l,) (V, v, h,) isthe equalizer of (g,) and (g,).

Remark. Notice that this result, specifically PS 2), isfalse for finer topologies.

Proposition 8.6. IfF is a sheaf on Affet/X, then fet F is a sheaf.

Proof. It can be proven similarly to Proposition 6.7.

Theorem 8.7. Let f: X Y be a 1-morphism of geometric Deligne-Mumfordstacks, there is an associated morphism of topos

fet : Xet Yet

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Proof. Let G Yet, we denote by f1et G the sheaf associated to the presheaffpG. This, together with Proposition 8.6 provides an adjunction

Xetf1etfet

Yet

The pair (fet ,f1et

) defines a morphism of topoi, since f1et

is exact because sois fp and the sheafification functor.

Proposition 8.8. Let f: X Y and g : Y Z be 1-morphisms of geometricDeligne-Mumford stacks. It holds that (g f)et

=get fet

Proof. The proof is mutatis mutandis the same as Proposition 6.9.

Corollary 8.9. In the previous situation (g f)1et

f1et

g1et

.

Proof. It is is a consequence of the previous proposition by adjunction.

8.10. Let f: X Y be a 1-morphism of geometric Deligne-Mumford stacks.The ring homomorphism f#

et: OY fOX induces a canonical morphism of

ringed topoi(fet,f

et# ) : (Xet,OX) (Yet,OY)

where fet# : f1OY OX is the morphism adjoint to f#et. Now we can define

the functor fet

: OY-ModOX-Mod by the usual formula

fetF :=OX f1et OY f1et F.

It is left adjoint to fet : OX-ModOY-Mod.

Proposition 8.11. In the previous situation, we have

(i) ifGQcoet(Y) then fetGQcoet(X), and(ii) ifFQcoet(X) then fet FQcoet(Y).

Proof. This can be shown in a similar way to Propositions 6.16 and 6.17.

Corollary 8.12. There is a pair of adjoint functors

Qcoet(X)fet

fet

Qcoet(Y).

Proof. Combine the previous proposition with 8.10.

For X a geometric Deligne-Mumford stack we denote by rX : Xfppf Xet therestriction functor.

Proposition 8.13. Let f: X Y be a 1-morphism of geometric Deligne-Mum-ford stacks. There is an isomorphism of functors from Xfppf to Yet rYf = fet rX.

Proof. As p is an affine morphism, then rXp = pet rX. In general, let F Xfppf. Recall that we have an equalizer in Xfppf

FF

TFTFTFT2F, (8.13.1)

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44 L. ALONSO, A. JEREMAS, M. PREZ, AND

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