aa r X i v : . [ m a t h . AG ] S e p SHEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION
FABIO TONINI
Abstract.
We introduce “sheafification” functors from categories of (lax monoidal) linearfunctors to categories of quasi-coherent sheaves (of algebras) of stacks. They generalize thehomogeneous sheafification of graded modules for projective schemes and have applicationsin the theory of non-abelian Galois covers and of Cox rings and homogeneous sheafificationfunctors. Moreover, using this theory, we prove a non-neutral form of Tannaka’s reconstruction,extending the classical correspondence between torsors and strong monoidal functors.
Introduction If G is an affine group scheme over a field k , classical Tannaka’s reconstruction problemconsists in reconstructing the group G from Rep G k , its category of finite representations: if F : Rep G k −→ Vect k is the forgetful functor then G is canonically isomorphic to the sheaf ofautomorphisms of F (opportunely defined, see [DM82, Proposition 2.8]). More generally one canrecover the stack B G of G -torsors by looking at fiber functors. Given a ring A denote by Loc A the category of locally free sheaves of finite rank over A , that is finitely generated projective A -modules. If SMon Gk is the stack over k whose fiber over a k -algebra A is the category of k -linear,strong monoidal and exact functors Γ :
Rep G k −→ Loc A then the functorB G −→ SMon Gk , ( Spec A s −−→ B G ) ( s ∗| Rep G k : Rep G k −→ Loc A ) ( ⋆ )is an equivalence of categories (see [DM82, Theorem 3.2]). Here Rep G k is thought of as thecategory of locally free sheaves of finite rank on B G .The above functor can be defined in a way more general context. Let us introduce somenotations and definitions. We fix a base commutative ring R and a category fibered in groupoids X over R . We say that X is pseudo-algebraic (resp. quasi-compact ) if there exists a scheme (resp.affine scheme) X and a map X −→ X representable by fpqc covering of algebraic spaces. Wedenote by QCoh X and Loc X the categories of quasi-coherent sheaves and locally free sheavesof finite rank respectively (see Section 1). Given a full subcategory C of QCoh X we say that C generates QCoh X if all quasi-coherent sheaves on X are quotient of a (possibly infinite) directsum of sheaves in C . We say that X satisfies (or has) the resolution property if Loc X generatesQCoh X .Given a monoidal and additive full subcategory C of Loc X and a category fibered in groupoids Y over R define Fib X , C ( Y ) as the category of R -linear and strong monoidal functors Γ :
C −→
Loc Y which are exact on right exact sequences in C (in the ambient abelian category QCoh X ).Denote also by Fib X , C the stack over R whose fiber over an R -algebra A is Fib X , C ( Spec A ) andby P C the functor P C : X −→
Fib X , C , ( Spec A s −−→ X ) ( s ∗|C : C −→
Loc A ) The functor ( ⋆ ) is obtained by taking R = k , X = B G and C = Loc X . We prove the followingnon-neutral form of Tannaka’s reconstruction. Theorem (5.3, 5.4) . Let X be a quasi-compact stack over R for the fpqc topology with quasi-affine diagonal and C ⊆
Loc X be a full, additive and monoidal subcategory with duals generating QCoh X .If Γ :
C −→
Mod A, where A is an R -algebra, is an R -linear, contravariant and strong monoidalfunctor such that Γ , as well as Γ ⊗ A k for all geometric points Spec k −→ Spec A , is left exacton right exact sequences in C then there exists Spec A s −−→ X such that Γ ≃ ( s ∗ ) ∨ .The functor P C : X −→
Fib X , C is an equivalence of stacks and, if Y is a category fibered ingroupoids, the functor Hom ( Y , X ) −→ Fib X , C ( Y ) , ( Y f −−→ X ) f ∗|C : C −→
Loc ( Y ) is an equivalence of categories. Notice that the two conclusions in the last statement are equivalent. In the case C = Loc X ,the functor P Loc ( X ) has already been proved to be an equivalence in the neutral case, that is X = B R G , where G is a flat and affine group scheme over R (see [Bro13, Theorem 1.2], where R is a Dedekind domain, and [Sch13, Theorem 1.3.2] for general rings R ), for particular quotientstacks over a field (see [Sav06] and 5.12) and for quasi-compact and quasi-separated schemes (see[BC12, Proposition 1.8]). We also show an almost converse of Theorem above: Theorem. [5.7] Let X be a quasi-compact category fibered in groupoids over R admitting asurjective (on equivalence classes of geometric points) map X −→ X from a scheme whoseconnected components are open (e.g. a connected or Noetherian algebraic stack) and let C ⊆
Loc X be a full monoidal subcategory with duals such that Sym n E ∈ C for n ∈ N and E ∈ C if E has local rank not invertible in R (e.g. C = Loc ( X ) or C consists of invertible sheaves). ThenFib X , C is a quasi-compact stack in groupoids for the fpqc topology over R with affine diagonaland with a collection of tautological locally free sheaves {G E } E∈C generating
QCoh ( Fib X , C ) andsuch that P ∗C G E ≃ E . For instance it follows that
X −→
Fib X , Loc ( X ) is universal among the maps w : X −→ Y where Y is a quasi-compact stack over R with quasi-affine diagonal and the resolution property.There are also variants of theory above where Loc ( − ) is replaced by Coh ( − ) or QCoh ( − ) or thederived category D ( − ) (see [Lur04, Sch12, BC12, Bra14, Bha14]). Although not explicitly statedelsewhere, for stacks with the resolution property and with affine diagonal (which is automaticin the algebraic case, see [Tot04]) those results and the fact that P Loc ( X ) is an equivalence canbe proved to be equivalent: one can pass from quasi-coherent sheaves to locally free sheaves viadualizable objects and, for the converse, extend functors from Loc ( − ) to QCoh ( − ) following theproof of [Bha14, Corollary 3.2]. We complete this picture by showing that in general the resolutionproperty implies the affineness of the diagonal (see 5.13). One of the ingredients in the proof isthe classification of quasi-compact stacks whose quasi-coherent sheaves are generated by globalsections, called pseudo-affine. In the algebraic case those coincide with quasi-affine schemes(see [Gro13, Proposition 3.1]), while, in general, we prove they are (arbitrary) intersection ofquasi-compact open subschemes (thought of as sheaves) of affine schemes (see 5.6).The proof of Tannaka’s reconstruction we present does not reduce to the case of quasi-coherentsheaves as explained above but it follows a different path. It is obtained by developing a theoryof sheafification functors which I think is interesting on its own and it is the heart of the paper.In what follows A will denote an R -algebra, X a category fibered in groupoids over R and C afull subcategory of QCoh X . The idea is simple: if s : Spec A −→ X is a map (say quasi-affineso that s ∗ preserves quasi-coherency) then we have natural isomorphisms of A -modules s ∗ E ≃
Hom A ( s ∗ E ∨ , A ) ≃ Hom X ( E ∨ , s ∗ O Spec A ) HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 3
By passing from covariant functors to contravariant ones we can always define Ω F : C −→
Mod A, Ω FE = Hom X ( E , F ) for F ∈
QCoh ( X × A ) which are R -linear contravariant functors. If L R ( C , A ) is the category of contravariant R -linearfunctors C −→
Mod A we obtain a functor Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) Quite surprisingly (since we started from group schemes as motivation) we recover also this wellknown situation (see 2.2 for details). If X is a quasi-projective and quasi-compact scheme over R with very ample invertible sheaf O X (1) consider C X = {O X ( n ) } n ∈ Z , set S X for the homogeneouscoordinate ring of ( X, O X (1)) and use GMod ( − ) to denote the category of graded modules. Wehave that L R ( C X , A ) is equivalent to GMod ( S X ⊗ R A ) and Ω ∗ corresponds to Γ ∗ : QCoh ( X × A ) −→ GMod ( S X ⊗ R A ) , Γ ∗ ( F ) = M n ∈ Z H ( X, F ( n )) It is a classical result, at least over a field, that the above functor is fully faithful. Thus in generalwe can ask under what conditions the functor Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) is fully faithfulas well. Another property that the functor Γ ∗ has is the existence of a left adjoint, namely thehomogeneous sheafification functor e − : GMod ( S X ⊗ R A ) −→ QCoh ( X × A ) . It turns out thatalso Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) has a left adjoint F ∗ , C : L R ( C , A ) −→ QCoh ( X × A ) when C is essentially small: by analogy we call this functor a sheafification functor.In the context of Tannaka’s reconstruction we also have that sheaves of algebras correspondto (lax) monoidal functors Loc G R −→ Mod A . This is true in general. If C is a monoidalsubcategory of QCoh X , ML R ( C , A ) denotes the category of R -linear, contravariant and monoidalfunctors C −→
Mod A and QAlg ( X × A ) the category of quasi-coherent sheaves of algebras on X then Ω ∗ extends to a functor Ω ∗ : QAlg ( X × A ) −→ ML R ( C , A ) and, if C is essentially small, F ∗ , C extends to A ∗ , C : ML R ( C , A ) −→ QAlg ( X × A ) , which is still a left adjoint of Ω ∗ .Coming back to the fully faithfulness of Ω ∗ we prove the following. Theorem (3.12,3.18) . Let X be a pseudo-algebraic category fibered in groupoids over R and C ⊆
QCoh X be a full subcategory generating QCoh X . Then the functor Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) is fully faithful and, if C is essentially small, then F ∗ , C : L R ( C , A ) −→ QCoh ( X × A ) isexact and the natural map G −→ F Ω G , C is an isomorphism. If X is quasi-projective then it is a classical fact that C X = {O X ( n ) } n ∈ Z generates QCoh X and thus we recover the classical properties of Γ ∗ : QCoh ( X × A ) −→ GMod ( S X ⊗ R A ) . Theoremabove when C consists of a single object is a rephrasing of classical Gabriel-Popescu’s theoremfor the category QCoh X (see 3.19). When C is monoidal and generates QCoh X we also havethat Ω ∗ : QAlg ( X × A ) −→ ML R ( C , A ) is fully faithful (see 3.34).The second problem we address is to describe the essential image of Ω ∗ . The main idea behindthis description is that Hom X ( − , F ) for F ∈
QCoh X is a left exact functor. Since the domain C of the functors Ω F is not abelian we need an ad hoc definition of exactness. A test sequence in C is an exact sequence T ∗ : M k ∈ K E k α −−→ M i ∈ I E i −→ E −→ where E , E i , E k ∈ C and I, J are setssuch that α ( E k ) is contained in a finite sum for all k ∈ K . A test sequence is called finite if K and I are finite sets. Given Γ ∈ L R ( C , A ) we will say that Γ is exact on a test sequence T ∗ in C if the complex of A -modules (see 3.9) −→ Γ E −→ Y i ∈ I Γ E i −→ Y k ∈ K Γ E k HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 4 is exact. We denote by Lex R ( C , A ) the full subcategory of L R ( C , A ) of functors which are exactan all test sequences. We have the following: Theorem (3.18, 3.26) . Let X be a pseudo-algebraic category fibered in groupoids over R and C ⊆
QCoh X be a full subcategory generating QCoh X . Then Lex R ( C , A ) is the essential imageof the (fully faithful) functor Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) . If X is quasi-compact and allsheaves in C are finitely presented then Lex R ( C , A ) is the subcategory of L R ( C , A ) of functorswhich are exact on finite test sequences. In particular, when C is essentially small, Ω ∗ : QCoh ( X × A ) −→ Lex R ( C , A ) and F ∗ , C : Lex R ( C , A ) −→ QCoh ( X × A ) are quasi-inverses of each other. The two theorems above apply in the followingsituations (see 3.29, 3.30 and 3.31): if C = QCoh X then Lex R ( QCoh X , A ) is the category ofcontraviant, R -linear and left exact functors QCoh X −→
Mod A which transform direct sumsinto products, if C = Coh ( X ) (resp. C = Loc ( X ) ) and X is a notherian algebraic stack (resp. X is quasi-compact and has the resolution property) then Lex R ( C , A ) is the category of contraviant, R -linear and left exact functors C −→
Mod A .When C is essentially small there is another cohomological characterization of the functorsin Lex R ( C , A ) . A collection of maps U = {E i −→ E} in C is called jointly surjective if the map L i ∈ I E i −→ E is surjective. Given such a collection U we set ∆ U = Im ( L i ∈ I Ω E i −→ Ω E ) ∈ L R ( C , R ) . Denote by C ⊕ the subcategory of QCoh X consisting of all possible finite direct sumsof sheaves in C . We have: Theorem (3.24, 3.26 and 3.28) . Let X be a pseudo-algebraic category fibered in groupoids over R and C ⊆
QCoh X be a full and essentially small subcategory generating QCoh X . Then Lex R ( C , A ) is the full subcategory of L R ( C , A ) of functors Γ satisfying Hom L R ( C ,R ) (Ω E / ∆ U , Γ) =
Ext L R ( C ,R ) (Ω E / ∆ U , Γ) = 0 for all jointly surjective collections of maps U = {E i −→ E} i ∈ I in C . If X is quasi-compact andthe sheaves in C are finitely presented we can consider only finite collections U .We have Lex R ( C ⊕ , A ) ≃ Lex R ( C , A ) via the restriction C −→ C ⊕ and, if C is additive and J is the smallest Grothendieck topology on C containing the sieves ∆ U for all jointly surjectivecollections U = {E i −→ E} i ∈ I in C , then Lex R ( C , A ) coincides with the category of sheaves of A -modules C op −→ Mod A on the site ( C , J ) which are R -linear. Besides Tannaka’s reconstruction problem, theory above has two other applications. The first,which is also the original motivation, is the theory of Galois cover. More precisely in my Ph.D.thesis [Ton13b] I have worked out theory above in the case X = B G and C = Loc X , where G isa finite, flat and finitely presented group scheme over R . Notice that B G satisfies the resolutionproperty in this case (see 4.3). The proof presented in [Ton13b] makes use of representationtheory and can not be generalized to arbitrary categories fibered in groupoids. The goal was tolook at Galois covers with group G as particular monoidal functors, as G -torsors can be thoughtof as particular strong monoidal functors, and the motivation was the study of non-abelian Galoiscovers, where a direct approach as in the abelian case (see [Ton13a]) fails due to the complexityof the representation theory.A second application is to the theory of Cox rings and homogeneous sheafifications. The ideais to consider C H = {L} L∈ H ⊆ Loc X where H is a subgroup of Pic X . As in the projective casewe have a homogeneous coordinate ring S H = M L∈ H H ( X , L ) HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 5 (opportunely defined), L R ( C H , A ) is equivalent to GMod ( S H ⊗ R A ) , the category of H -graded ( S H ⊗ R A ) -modules, Ω ∗ corresponds to Γ ∗ : QCoh ( X × A ) −→ GMod ( S H ⊗ R A ) , Γ ∗ ( F ) = M L∈ H H ( X , F ⊗ L ) and its adjoint F ∗ , C H behaves like a homogeneous sheafification. Moreover in more concretegeometric situations, e.g. when X is a normal variety, there are analogous constructions forreflexive sheaves of rank . We expect that this theory covers all known cases where Γ ∗ is provedto be fully faithful (see for instance [CLS11, Appendix of Chapter 6] and [Kaj98, Section 2]).Applications above are not described in the present paper, but, hopefully, they will be subjectsof future ones.The outline of the paper is the following. The first section introduces the notion and the basicproperties of quasi-coherent sheaves on fibered categories, while the second one is a general studyof sheafification functors. In the third section we study the fully faithfulness and the essentialimage of the functor Ω ∗ : QCoh ( X × A ) −→ L R ( C , A ) . In the fourth section we rewrite theresults obtained in the case of the stack of G -torsor in terms of the representation theory of G and finally, in the last section, we prove the non-neutral Tannaka’s reconstruction. Notation
In this paper we work over a base commutative, associative ring R with unity. If not statedotherwise a fiber category will be a category fibered in groupoids over Aff /R , the category ofaffine schemes over Spec R , or, equivalently, the opposite of the category of R -algebras. A schemeor an algebraic space X over Spec R will be thought of as the fibered category of maps from anaffine scheme to X , denoted by Aff /X . A map or morphism of fibered categories is a functorover Aff /R . Recall that by the -Yoneda lemma objects of a fibered category X can be thoughtof as maps T −→ X from an affine scheme. An fpqc stack will be a stack for the fpqc topology.A map f : X ′ −→ X of fibered categories is called representable if for all maps T −→ X froman affine scheme (or an algebraic space) the fiber product T × X X ′ is (equivalent to) an algebraicspace.Given a flat and affine group scheme G over T we denote by B R G the stack of G -torsors forthe fpqc topology, which is an fpqc stack with affine diagonal. When G −→ Spec R is finitelypresented (resp. smooth) then B R G coincides with the stack of G -torsors for the fppf (resp.étale) topology.By a “subcategory” of a given category we mean a “full subcategory” if not stated otherwise. Acknowledgements
I would like to thank Jarod Alper, David Rydh, Daniel Schäppi, Mattia Talpo and AngeloVistoli for the useful conversations I had with them and all the suggestions they gave me.1.
Preliminaries on sheaves and fibered categories
Let π : X −→ Aff /R be a fibered category. There is a functor of rings O X : X op −→ ( Sets ) defined by O X ( ξ ) = H ( O π ( ξ ) ) , so that ( X , O X ) is a ringed category. A presheaf of O X -moduleson X is a functor F : X op −→ ( Sets ) together with an H ( O π ( ξ ) ) -module structure on F ( ξ ) for all ξ ∈ X such that, if ξ −→ η is a map on X , then the map F ( η ) −→ F ( ξ ) is H ( O π ( η ) ) -linear. Morphism of presheaves of O X -modules are natural transformations respecting the modulestructures. We denote by Mod O X the category of presheaves of O X -modules. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 6
Definition 1.1.
A quasi-coherent sheaf over X is a presheaf of O X -modules such that for allmaps ξ −→ η in X the induced map F ( η ) ⊗ H ( O π ( η ) ) H ( O π ( ξ ) ) −→ F ( ξ ) is an isomorphism. We denote by QCoh X the full subcategory of Mod O X of quasi-coherentsheaves.Notice that by fpqc descent of modules a quasi-coherent sheaf is a sheaf for the fpqc topologyof X .If f : Y −→ X is a morphism of fibered categories and
F ∈
Mod O X we define f ∗ F = F ◦ f : Y op f −−→ X op F −−→ ( Sets ) . This association defines a functor f ∗ : Mod O X −→ Mod O Y ,called the pull-back functor, and restricts to a functor f ∗ : QCoh
X −→
QCoh Y . Notice that f ∗ O X = O Y tautologically.The category Mod O X is an abelian category and cokernels of maps between quasi-coherentsheaves are again quasi-coherent and thus cokernels in QCoh X . Notice that, since we haverestricted our fiber categories to affine schemes, a map f : F −→ G of quasi-coherent sheavesis an epimorphism if and only if it is pointwise surjective, that is surjective in Mod O X . Thecategory QCoh X is R -linear but it is unclear if it is abelian. Moreover kernels in Mod O X ofmaps between quasi-coherent sheaves are almost never quasi-coherent, essentially because pull-backs are not left exact. There is a natural condition on X which allows us to prove that QCoh X is an R -linear abelian category. Definition 1.2. An fpqc atlas (or simply atlas ) of a fibered category X is a representable fpqccovering X −→ X from a scheme. A fiber category is called pseudo-algebraic if it has an atlas,it is called quasi-compact if it has an atlas from an affine scheme.Let f : Y −→ X be a morphism of fibered categories. The map f is called pseudo-algebraic (resp. quasi-compact ) if for all maps T −→ X from a scheme (resp. quasi-compact scheme) thefiber product T × X Y is pseudo-algebraic (resp. quasi-compact). It is called quasi-separated ifthe diagonal Y −→ Y × X Y is quasi-compact.If X is pseudo-algebraic then the diagonal ∆ X : X −→ X × R X is not necessarily representable.It is unclear whether this is true or not if X is a fpqc stack, because it is not known if algebraicspaces satisfy effective descent along fpqc coverings.Let f : Y −→ X be a map of fibered categories. If X and f are pseudo-algebraic then Y ispseudo-algebraic. If Y is pseudo-algebraic and ∆ X is representable then f is pseudo-algebraic.If C is a (not full) subcategory of X one can analogously define presheaves of ( O X ) | C -modulesand quasi-coherent sheaves on C just replacing all occurrences of X with C . We denote byMod ( O X ) | C and QCoh C the resulting categories. Definition 1.3.
We define X fl (resp. X sm , X et ) as the (not full) subcategory of X of objects ξ : Spec B −→ X which are representable and flat (resp. smooth, étale) and the arrows aremorphisms in X whose underlying map of affine schemes is flat (resp. smooth, étale).If X −→ X is an fpqc atlas then by definition R = X × X X is an algebraic space and the twoprojections R ⇒ X extends to a groupoid in algebraic spaces. We denote by QCoh ( R ⇒ X ) thecategory of quasi-coherent sheaves on R ⇒ X (see [SP014, Tag 0440]). By standard argumentsof fpqc descent for modules we have: Proposition 1.4. If X admits an fpqc (resp. smooth, étale) atlas then the restriction QCoh
X −→
QCoh X fl (resp. QCoh X sm , QCoh X et ) is an equivalence of categories. If f : X −→ X is anfpqc atlas then f ∗ : QCoh
X −→
QCoh X is faithful and it induces an equivalence QCoh
X −→
QCoh ( R ⇒ X ) . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 7
We see that if X is pseudo-algebraic then QCoh X is equivalent to an R -linear abelian category,namely QCoh ( R ⇒ X ) . Moreover if α : F −→ G is a map of quasi-coherent sheaves then Ker ( α ) is defined by taking Ker ( α |X fl ) ∈ QCoh X fl , which is just given by Ker ( α |X fl )( Spec B ξ −−→ X ) = Ker ( α ( ξ ) : F ( ξ ) −→ G ( ξ )) for ξ ∈ X fl , and then extending it to the whole X . If X is an algebraicstack or a scheme we see that QCoh X is equivalent to the usual category of quasi-coherentsheaves via an R -linear and exact functor. If it is given a subcategory D of QCoh X , an exactsequence of sheaves in D will always be an exact sequence in QCoh X of sheaves belonging to D .We now deal with the problem of defining a right adjoint of a pull-back functor. Given F ∈
Mod O X we define the global section F ( X ) = Hom ( O X , F ) of F , also denoted by H ( X , F ) or simply H ( F ) , which is an O X ( X ) -module. More generally given a map of fibered categories g : Z −→ X we define F ( Z ) = ( g ∗ F )( Z ) . If Z = Spec B is affine we will often write F ( B ) instead of F ( Spec B ) .Let f : Y −→ X be a map of fibered categories. Given
G ∈
Mod O Y and an object ξ : T −→ X of X we define ( f p G )( ξ ) = G ( T × X Y ) Given another object ξ ′ : T ′ −→ X and a morphism β : ξ ′ −→ ξ in X there is an inducedmorphism ( f p G )( ξ ) −→ ( f p G )( ξ ′ ) . This data define a functor f p : Mod O Y −→ Mod O X and wehave: Proposition 1.5.
Let f : Y −→ X be a map of fibered categories. Then f p is a right adjoint of f ∗ and, if Y ′ YX ′ X g ′ g ff ′ is a -cartesian diagram of fibered categories, there is an isomorphism of functors g ∗ f p −→ f ′ p g ′∗ : Mod O Y −→ Mod O X ′ If f is affine then f p ( QCoh Y ) ⊆ QCoh X and ( f p ) | QCoh Y : QCoh
Y −→
QCoh X is right adjointto f ∗ : QCoh
X −→
QCoh Y .Proof. The adjunction between f ∗ and f p can be found in [SP014, Tag 00XF]. With notation fromthis reference, we have that fη I ≃ T × X Y for an object η : T −→ X . Moreover if F ∈
Mod O Y then F ( T × X Y ) is the limit of F | T × X Y over the whole T × X Y . Thus what is denoted by p f is easily seen to be equivalent to our f p if we take limits of R -modules and not of sets. Theisomorphism for the base change is tautological. For the last claim we can assume that X isan affine scheme in which case the result follows because (usual) push-forwards commutes witharbitrary base changes. (cid:3) In general f p does not preserve quasi-coherent sheaves, even if f is a proper map of schemes.To get a right adjoint of pullback we have to require more. Definition 1.6.
A pseudo-algebraic map f : Y −→ X of fibered categories is called flat if givenan object ξ : T −→ X of X and an atlas V −→ T × X Y the resulting map V −→ T is flat. Proposition 1.7.
Let f : Y −→ X be a quasi-compact and quasi-separated map of pseudo-algebraic fibered categories. Then the composition
QCoh Y f p −−→ Mod O X −→ Mod ( O X ) |X fl has values in QCoh X fl . The induced map f ∗ : QCoh
Y −→
QCoh X is a right adjoint of HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 8 f ∗ : QCoh
X −→
QCoh Y . If Y ′ YX ′ X g ′ g ff ′ is a -cartesian diagram of fibered categories with X ′ pseudo-algebraic then Y ′ is pseudo-algebraic, f ′ is quasi-compact and quasi-separated and there is a natural transformation of functors g ∗ f ∗ −→ f ′∗ g ′∗ : QCoh
X −→
QCoh Y ′ which is an isomorphism if g is flat.Proof. Consider the -Cartesian diagram in the statement. The diagonal of f ′ is quasi-compactbecause it is base change of the diagonal of f . To see that f p ( F ) |X fl is quasi-coherent for F ∈
QCoh Y , we can assume X = Spec B affine and that Y is quasi-compact with quasi-compactdiagonal. If U = Spec A −→ Y is a fpqc atlas, it follows that R = U × Y U is a quasi-compactalgebraic space. By covering R by finitely many affine schemes Spec A i we can write F ( Y ) askernel of a map F ( A ) −→ ⊕ i F ( A i ) . If we base change along a flat map B −→ B ′ it is now easyto see that F ( Y × B B ′ ) ≃ F ( Y ) ⊗ B B ′ , as required.To define the natural transformation α : g ∗ f ∗ −→ f ′∗ g ′∗ notice that there is a natural map f ∗ F −→ f p F which extends the identity on X fl . Applying g ∗ we get g ∗ f ∗ F −→ g ∗ f p F ≃ f ′ p g ′∗ F and then, restricting to X ′ fl , a map ( g ∗ f ∗ F ) |X ′ fl −→ ( f ′∗ g ′∗ F ) |X ′ fl . Since both sides are in QCoh X ′ fl this map uniquely extends to a natural transformation α as required. Finally assume that g isflat and let ξ : Spec B −→ X ′ ∈ X ′ fl . If the composition Spec B −→ X is in X fl then one caneasily check that α ( ξ ) is an isomorphism. Otherwise, by definition of flatness, there exists anfpqc covering Spec B ′ −→ Spec B whose composition ξ ′ : Spec B ′ −→ X ′ satisfies the previouscondition. Since α ( ξ ) ⊗ B B ′ ≃ α ( ξ ′ ) we get the desired result. (cid:3) Remark . There are set-theoretic problems in considering global sections of presheaves andtherefore push-forwards, because Mod O X is in general not locally small. The common way tosolve this problem is to use Grothendieck universes. Take a universe U and define rings inside U , so that Aff /R is small (with respect to a bigger universe). Fibered categories should thenbe required to be small too. In this situation it is easy to show that Mod O X is locally smalland therefore global sections and push-forwards are well defined. With this approach we haveto be careful in considering (big) rings defined starting from some F ∈
Mod O X : for instanceSpec O X ( X ) is in general not an object of Aff /R .Notice that global sections and pushforwards of quasi-coherent sheaves are always well definedfor a pseudo-algebraic fibered category and a pseudo-algebraic map respectively. The reason isthat if F ∈
QCoh X and p : X −→ X is a fpqc atlas then F ( X ) −→ ( p ∗ F )( X ) is injective andthus F ( X ) is a set.In the rest of the paper we will not be concerned about those set-theoretic problems. Definition 1.9. If A is an R -algebra and F ∈
Mod O X then a compatible A -module structure on F is the data of A -module structures on F ( ξ ) commuting with the H ( O π ( ξ ) ) -module structureon F ( ξ ) for all ξ ∈ X and such that, for all ξ −→ η in X , the map F ( η ) −→ F ( ξ ) is A -linear. Wedefine QCoh A X as the category of quasi-coherent sheaves over X with an A -module structure.We also define X A as the fiber product Spec A × R X .Notice that if Y g −−→ X is a map of fibered categories and F is a presheaf of O X -modules withan A -module structure then g ∗ F inherits an A -module structure. In particular g ∗ : QCoh
X −→
QCoh Y extends to a functor QCoh A X −→
QCoh A Y . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 9
Proposition 1.10.
Let A be an R -algebra. Then the push-forward map QCoh X A −→ QCoh X extends naturally to an equivalence QCoh X A −→ QCoh A X .Proof. The result is very simple if X is an affine scheme. In general, if we set g : X A −→ X forthe projection and we consider G ∈
QCoh X A , then g ∗ G ∈
QCoh X and it inherits an A -modulestructure from the action of A on G . Therefore g ∗ G ∈
QCoh A X . If h : Spec B −→ X is a mapconsider the diagramsSpec ( B ⊗ R A ) X A QCoh X A QCoh Spec ( B ⊗ R A ) Spec B X QCoh A X QCoh A Spec B h ′ gh g ′∗ g ∗ g ′ h ′∗ h ∗ The second diagram is -commutative and the last vertical map is an equivalence. Using thosediagrams it is easy to define a quasi-inverse QCoh A X −→
QCoh X A of g ∗ . (cid:3) We will almost always regard quasi-coherent sheaves over X A as objects of QCoh A X . Remark . If F , G ∈
QCoh A X then F ⊗ O X G does not correspond to the tensor product inQCoh X A : F ⊗ O X G has two distinct A -module structures. Under the equivalence QCoh X A −→ QCoh A X the tensor product of F and G , that we will denote by F ⊗ O X A G , is given by U ( U ) ⊗ H ( O U ) G ( U ) / h ax ⊗ y − x ⊗ ay | x ∈ F ( U ) , y ∈ G ( U ) i Definition 1.12.
A locally free sheaf E (of rank n ) over X is a quasi-coherent sheaf suchthat E ( Spec B −→ X ) is a finitely generated projective B -module (of rank n ) for all mapsSpec B −→ X . We denote by Loc X the subcategory of QCoh X of locally free sheaves.We will say that a fiber category X has the resolution property if Loc X generates QCoh X .2. Sheafification functors.
In this section we define and describe particular functors that generalize sheafification functorsfor affine schemes or projective schemes. The idea is to interpret the category of modules orgraded modules respectively as a category of R -linear functors. More precisely: Definition 2.1.
Given a fibered category X over a ring R , an R -algebra A and a subcategory D of QCoh X we define L R ( D , A ) as the category of contravariant R -linear functors Γ :
D −→
Mod A and natural transformations as arrows. We define a functor Ω ∗ : QCoh A X −→ L R ( D , A ) by Ω F− = Hom X ( − , F ) : D −→
Mod A The functor Ω ∗ is called the Yoneda functor associated with D . A left adjoint of Ω ∗ is called a sheafification functor associated with D . If F ∈
QCoh A X we will call Ω F the Yoneda functorassociated with F . Example 2.2.
The analogy with the sheafification functor associated with X = P nR or anyquasi-projective and quasi-compact scheme over R is the following. If C = {O X ( n ) } n ∈ Z then with Γ ∈ L R ( C , R ) we can associate the Z -graded R -module M = L n ∈ Z Γ O X ( − n ) . The functorial prop-erties of Γ allow us to define a structure of graded S -module on M , where S = L n ∈ Z H ( O X ( n )) isthe homogeneous coordinate ring of X . This associations extends to an equivalence of categoriesbetween L R ( C , R ) and the category of graded S -modules. The functor Ω ∗ corresponds to thefunctor Γ ∗ which carries a quasi-coherent sheaf F on X to the graded S -module L n ∈ Z H ( F ( n )) .Finally the sheafification functor from the graded S -modules to QCoh X is left adjoint to Γ ∗ .Let us fix an R -algebra A and a fibered category π : X −→ Aff /R . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 10
Sheafifying R -linear functors. In this section we want to explicitly describe sheafificationfunctors for small subcategories of QCoh X . In particular we fix a small (and non empty)subcategory C of QCoh X .In the construction of the sheafification functors we will make use of the coend constructionin the settings of categories enriched over categories of modules over a ring. The general theorysimplifies considerably in this context and we will also apply such construction only in particularcases. In the following remark we collect all the properties we will need. Remark . Let Y be a fibered category over R , F : C −→
QCoh Y be an R -linear functor and Γ ∈ L R ( C , A ) . The coend of the R -linear functor Γ − ⊗ R F − : C op ⊗ C −→ QCoh A Y , denoted by ˆ E∈C Γ E ⊗ R F E ∈ QCoh A Y is the cokernel of the map M E u −−→ E (Γ u ⊗ id F E − id Γ E ⊗ F u ) : M E u −−→ E Γ E ⊗ R F E −→ M E∈C Γ E ⊗ R F E Moreover it comes equipped with an A -linear natural isomorphismHom QCoh A Y ( ˆ E∈C Γ E ⊗ R F E , H ) α −−→ Hom L R ( C ,A ) (Γ , Hom Y ( F − , H )) for H ∈
QCoh A Y given by α ( ˆ E∈C Γ E ⊗ R F E u −−→ H )( x ) = u ◦ p E ( x ⊗ − ) : F E −→ H for E ∈ C , x ∈ Γ E where p E : Γ E ⊗ R F E −→ ´ E∈C Γ E ⊗ R F E for E ∈ C are the structure morphisms. Its inverse isuniquely determined by the expression α − (Γ v −−→ Hom Y ( F − , H )) ◦ p E : Γ E ⊗ R E −→ H , x ⊗ y v E ( x )( y ) for E ∈ C
Natural transformations F −→ F ′ and Γ −→ Γ ′ yields morphisms ´ E∈C Γ E ⊗ R F E −→ ´ E∈C Γ E ⊗ R F ′E and ´ E∈C Γ E ⊗ R F E −→ ´ E∈C Γ ′E ⊗ R F E respectively. Those can be defined either usingYoneda’s lemma and the above characterization of Hom ( ´ E∈C Γ E ⊗ R F E , − ) or directly using thedescription of ´ E∈C Γ E ⊗ R F E as a cokernel.All the above claims are standard in the theory of coend in the enriched settings (in our caseenriched by Mod R ), but, in this simplified context, it is elementary to prove them directly.We start by showing that C (and therefore any essentially small subcategory of QCoh X )admits a sheafification functor. Proposition 2.4.
The Yoneda functor Ω ∗ : QCoh A X −→ L R ( C , A ) has a left adjoint F − , C : L R ( C , A ) −→ QCoh A X given by F Γ , C : X op −→ ( Sets ) , X ∋ ξ ˆ E∈C Γ E ⊗ R E ( ξ ) ∈ Mod ( H ( O π ( ξ ) ) ⊗ R A ) where E ( ξ ) denotes the evaluation C −→
Mod H ( O π ( ξ ) ) of sheaves in ξ ∈ X . Alternatively F Γ , C = ˆ E∈C Γ E ⊗ R E ∈
QCoh A X where E denotes the inclusion C −→
QCoh X . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 11
Proof.
It is enough to apply 2.3 with Y = X and F : C −→
QCoh X the inclusion. Using thedescription of coend as cokernel one can check that the two functors defined in the statementare canonically isomorphic. (cid:3) Definition 2.5.
We denote by γ Γ : Γ − −→ Ω F Γ , C − = Hom X ( − , F Γ , C ) and δ G : F Ω G , C −→ G for Γ ∈ L R ( C , A ) and G ∈
QCoh A X the unit and the counit of the adjunction between Ω ∗ : QCoh A X −→ L R ( C , A ) and F ∗ , C : L R ( C , A ) −→ QCoh A X respectively.Given ξ ∈ X , E ∈ C , ψ ∈ E ( ξ ) and x ∈ Γ E we denote by x E ,ψ ∈ F Γ , C ( ξ ) the image of x ⊗ ψ under the map Γ E ⊗ R E ( ξ ) −→ F Γ , C ( ξ ) Proposition 2.6.
Let Γ ∈ L R ( C , A ) . The unit γ Γ : Γ − −→ Ω F Γ , C − = Hom X ( − , F Γ , C ) is given by Γ E x Hom X ( E , F Γ , C )( φ x E ,φ ) If G ∈
QCoh A X the counit δ G : F Ω G , C −→ G is given by F Ω G , C ( ξ ) ∋ x E ,ψ x ( ψ ) ∈ G ( ξ ) for E ∈ C , x ∈ Ω GE = Hom X ( E , G ) , ξ ∈ X , ψ ∈ E ( ξ ) Proof. If p E : Γ E ⊗ R E −→ F Γ , C for E ∈ C are the maps associated with the coend defining F Γ , C ,then p E ( x ⊗ ψ ) = x E ,ψ for x ∈ Γ E , ψ ∈ E ( ξ ) and ξ ∈ X . All the claims follows by a direct checkusing the explicit description of the isomorphismHom QCoh A X ( ˆ E∈C Γ E ⊗ R E , H ) α −−→ Hom L R ( C ,A ) (Γ , Ω H ) for H ∈
QCoh A X and its inverse given in 2.3. (cid:3) Given a map g : Y −→ X of fibered categories we want to express g ∗ F Γ , C ∈ QCoh A Y for Γ ∈ L R ( C , A ) as F g ∗ Γ ,g ∗ C for a suitable choice of g ∗ C ⊆
QCoh Y and g ∗ Γ ∈ L R ( g ∗ C , A ) . Definition 2.7.
Let Y be a fibered category, g : Y −→ X be a morphism and D be a subcategoryof QCoh X . We set g ∗ D for the subcategory of QCoh Y of sheaves g ∗ E for E ∈ D . If D ′ ⊆ QCoh Y is a subcategory containing g ∗ D we can define a restriction functorL R ( D ′ , A ) L R ( D , A )Γ Γ ◦ g ∗ g ∗ Proposition 2.8.
Let Y be a fibered category, g : Y −→ X be a morphism and D be a sub-category of QCoh Y such that g ∗ C ⊆ D . Then g ∗ : L R ( D , A ) −→ L R ( C , A ) has a left adjoint g ∗ : L R ( C , A ) −→ L R ( D , A ) and it is given by ( g ∗ Γ) G = ˆ E∈C Γ E ⊗ R Hom Y ( G , g ∗ E ) ∈ Mod A for Γ ∈ L R ( C , A ) , G ∈ D where
Hom Y ( G , g ∗ − ) is thought of as a functor C −→
Mod R . If Y = X and g = id X , so that C ⊆ D and ( id X ) ∗ : L R ( D , A ) −→ L R ( C , A ) is the restriction, then the unit Γ −→ ( id ∗X Γ) |C is anisomorphism for Γ ∈ L R ( C , A ) .Proof. Let Ω ∈ L R ( D , A ) . Applying 2.3 with F = Hom Y ( G , g ∗ − ) we get a bijection betweenHom A (( g ∗ Γ) G , Ω G ) and the set of A -linear natural transformations Γ E −→ Hom R ( Hom ( G , g ∗ E ) , Ω G ) for E ∈ C . A natural transformation g ∗ Γ −→ Ω corresponds to a collection γ of A -linear maps γ G , E : Γ E −→ Hom R ( Hom ( G , g ∗ E ) , Ω G ) natural in E ∈ C and such that γ G , E ( x )( φ ◦ u ) = Ω u ( γ G , E ( x )( φ )) for x ∈ Γ E , G φ −−→ g ∗ E , G u −−→ G ∈ D HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 12
Set µ γ ( x ) = γ g ∗ E , E ( x )( id g ∗ E ) ∈ Ω g ∗ E for E ∈ C , x ∈ Γ E . If γ is a collection as above it followsthat γ G , E ( x )( ψ ) = Ω ψ ( µ γ ( x )) for E ∈ C , G ∈ D , x ∈ Γ E and G ψ −−→ g ∗ E and that µ γ : Γ E −→ Ω g ∗ E = ( g ∗ Ω) E is A -linear and natural in E . Conversely, given a morphism µ : Γ −→ ( g ∗ Ω) inL R ( C , A ) we can always define a collection γ as above by setting γ G , E ( x )( ψ ) = Ω ψ ( µ ( x )) . It iseasy to check that γ induces a morphism g ∗ Γ −→ Ω and that the above constructions yields abijection Hom ( g ∗ Γ , Ω) ≃ Hom (Γ , g ∗ Ω) .Assume now Y = X and g = id X and let Γ ∈ L R ( C , A ) and E ∈ C . Denote by α : Γ −→ ( id ∗X Γ) |C the unit morphism. If p e E : Γ e E ⊗ Hom X ( E , e E ) −→ ( id ∗X Γ) E are the structure morphismsas in 2.3, then α E = p E ( − ⊗ id E ) . In particular, given H ∈
Mod A and using 2.3, the mapHom A ( α E , H ) : Hom A (( id ∗X Γ) E , H ) −→ Hom A (Γ E , H ) sends an A -linear natural transformation δ : Γ − −→ Hom R ( Hom X ( E , − ) , H ) to Γ E ∋ x δ ( x )( id E ) ∈ H . Since δ corresponds to an R -linear natural transformation Hom X ( E , − ) −→ Hom A (Γ − , H ) , by Yoneda lemma we see thatHom A ( α E , H ) is an isomorphism. (cid:3) The above proposition yields a natural extension of any Γ ∈ L R ( C , A ) to a functor Γ ex ∈ L R ( QCoh X , A ) . By abuse of notation we will denote them by the same symbol Γ . This meansthat if Γ ∈ L R ( C , A ) and G ∈
QCoh X then we can evaluate Γ on G , writing Γ G .Given a map g : Y −→ X we will denote by g ∗ : L R ( C , A ) −→ L R ( g ∗ C , A ) the left adjoint of therestriction L R ( g ∗ C , A ) −→ L R ( C , A ) . So, given Γ ∈ L R ( C , A ) , g ∗ Γ is a functor g ∗ C −→
Mod A but it also defines a functor QCoh Y −→
Mod A denoted, by our convention, by the same symbol.By 2.8 the functor g ∗ Γ :
QCoh
Y −→
Mod A coincides with the value of the left adjoint of therestriction L R ( QCoh Y , A ) −→ L R ( C , A ) . Remark . Given Γ ∈ L R ( C , A ) and E ∈ C we have R -linear morphisms of ringsH ( O X ) ≃ End X ( O X ) −→ End X ( E ) −→ End A (Γ E ) This defines a lifting of Γ to an R -linear functor Γ :
C −→
Mod ( H ( O X ) ⊗ R A ) and an equivalenceL R ( C , A ) −→ L R ( C , H ( O X ) ⊗ R A ) In particular, if g : Spec B −→ X is a map and Γ ∈ L R ( C , A ) then ( g ∗ Γ) B has a B ⊗ R A -modulestructure. By 2.4 and 2.8 there is a canonical A -linear isomorphism F Γ , C ( B ) ≃ ( g ∗ Γ) B and it is easy to see that it is also B -linear. Proposition 2.10.
Let g : Y −→ X be a morphism of fibered categories. Then there exists anisomorphism g ∗ F Γ , C ≃ F g ∗ Γ ,g ∗ C natural in Γ ∈ L R ( C , A ) , that is a -commutative diagram L R ( C , A ) QCoh A X L R ( g ∗ C , A ) QCoh A Y F − , C g ∗ g ∗ F − ,g ∗C Proof.
Let Γ ∈ L R ( C , A ) . Let also Spec B ξ −−→ Y be a map and N ∈ Mod B ⊗ R A . Denote by F : C −→
Mod B and G : g ∗ C −→
Mod B the functors obtained by taking sections over B . Inparticular F = G ◦ g ∗ . We have isomorphismsHom B ⊗ R A ( F g ∗ Γ ,g ∗ C ( B ) , N ) ≃ Hom L R ( g ∗ C ,A ) ( g ∗ Γ , Hom B ( G, N )) ≃ Hom L R ( C ,A ) (Γ , g ∗ Hom B ( G, N ))=
Hom L R ( C ,A ) (Γ , Hom B ( F, N )) ≃ Hom B ⊗ R A (( g ∗ F Γ , C )( B ) , N ) HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 13
In particular we get an isomorphism ( g ∗ F Γ , C )( B ) ≃ F g ∗ Γ ,g ∗ C ( B ) . By a direct check it followsthat this isomorphism is natural in Γ and it also extends as an isomorphism of functors g ∗ F Γ , C ≃F g ∗ Γ ,g ∗ C . (cid:3) Notice that, as soon as g ∗ : QCoh
X −→
QCoh Y has a right adjoint g ∗ : QCoh
Y −→
QCoh X (see 1.7), then g ∗ Ω G ≃ Ω g ∗ G for G ∈
QCoh A Y and the above proposition follows taking the leftadjoints. Remark . If A −→ A ′ is a morphism of R -algebras then we have pull-back functors L R ( C , A ) −→ L R ( C , A ′ ) and QCoh A X −→
QCoh A ′ X . The first one is obtained considering the tensor product − ⊗ A A ′ , while the second one corresponds to the pullback QCoh X A −→ QCoh X A ′ along theprojection X A ′ −→ X A . Alternatively, those functors are left adjoints to the restriction of scalarsL R ( C , A ′ ) −→ L R ( C , A ) and QCoh A ′ X −→
QCoh A X respectively. It is easy to see that in thisway we obtain two fpqc stacks (not in groupoids) L R ( C , − ) and QCoh − X over the category ofaffine R -schemes. Notice that the functor Ω ∗ : QCoh − X −→ L R ( C , − ) is not a morphism ofstacks because Hom X ( E , G ) ⊗ A A ′ Hom X ( E , G ⊗ A A ′ ) in general for E ∈ C and
G ∈
QCoh A X . Proposition 2.12.
The functor F ∗ , C : L R ( C , − ) −→ QCoh − X is a morphism of stacks.Proof. Given a morphism A −→ A ′ of R -algebras we have a -commutative diagramQCoh A ′ X L R ( C , A ′ ) QCoh A X L R ( C , A ) Ω ∗ Ω ∗ where the vertical arrows are obtained by restricting the scalars from A ′ to A . Using 2.11 andtaking the left adjoint functors of the functors in the diagram we exactly get the -commutativediagram expressing the fact that F ∗ , C preserves Cartesian arrows. (cid:3) We conclude this section by showing that, when considering sheafification functors F − , C , wecan always reduce problems to the case when C is an additive category. Moreover in this casethe sections of F − , C have a nice expression in terms of a direct limit. Definition 2.13.
Given a subcategory D of QCoh X we denote by D ⊕ the subcategory ofQCoh X whose objects are all finite direct sums of sheaves in D .Notice that if D is small then D ⊕ is small. Proposition 2.14.
The restriction L R ( C ⊕ , A ) −→ L R ( C , A ) and its left adjoint are inverses ofeach other. In particular if Γ ∈ L R ( C ⊕ , A ) then we have a canonical isomorphism F Γ , C ⊕ ≃ F Γ |C , C .Proof. Denote the restriction by α and its left adjoint by β . From 2.8 we know that αβ (Γ) ≃ Γ for Γ ∈ L R ( C , A ) . Conversely, if Γ ∈ L R ( C ⊕ , A ) , we have a canonical morphism γ : βα (Γ) −→ Γ . If E ∈ C the map γ E is easily seen to be an isomorphism and by additivity this also follows for γ . Forthe last claim it is enough to note that the composition QCoh A X Ω ∗ −−→ L R ( C ⊕ , A ) −→ L R ( C , A ) is exactly Ω ∗ : QCoh A X −→ L R ( C , A ) . (cid:3) Remark . If C is an R -linear and additive category and F, G : C −→ Mod A are R -linear(covariant or contravariant) functors then any natural transformation λ : F −→ G of functors ofsets is R -linear. Indeed by considering C op we can consider only covariant functors. In this caseit is easy to show that the maps λ X : F ( X ) −→ G ( X ) for X ∈ C are R -linear using functorialityon the map r id X : X −→ X for r ∈ R and pr , pr , pr + pr : X ⊕ X −→ X , where pr ∗ are theprojections. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 14
Definition 2.16.
Let Spec B −→ X be a map. We denote by J B, C the category of pairs ( E , ψ ) where E ∈ C ⊕ and ψ ∈ E ( B ) . Given Γ ∈ L R ( C , A ) we have a functor Γ : J B, C −→ Mod A givenby Γ E ,ψ = Γ E . Proposition 2.17.
Let
Spec B −→ X be a map and Γ ∈ L R ( C , A ) . The category J B, C is non-empty and for all ξ, ξ ′ ∈ J B, C there exists ξ ′′ ∈ J B, C and maps ξ ′′ −→ ξ , ξ ′′ −→ ξ ′ . The A -linearmaps Γ E ,ψ = Γ E −→ F Γ , C ⊕ ( B ) ≃ F Γ , C ( B ) , x x E ,ψ for ( E , ψ ) ∈ J B, C (see 2.5) induce an A -linear isomorphism lim −→ ( E ,ψ ) ∈ J op B, C Γ E ,ψ −→ F Γ , C ( B ) In particular all elements of F Γ , C ( B ) are of the form x E ,ψ for some E ∈ C ⊕ , x ∈ Γ E and ψ ∈ E ( B ) .The multiplication by b ∈ B on the first limit is induced by mapping Γ E ,ψ to Γ E ,bψ using id Γ E for ( E , ψ ) ∈ J B, C .Proof. By 2.14 we can assume C = C ⊕ . Denote by H and α : H −→ F Γ , C ( B ) the limit and themap in the statement respectively. The category J B, C is not empty because ( E , ∈ J B, C for all E ∈ C and the map α is well defined because, for all x ∈ Γ E and for all ( E , ψ ) u −−→ ( E , u ( ψ )) wehave x E ,u ( ψ ) = (Γ u ( x )) E ,ψ , by definition of F Γ , C ( B ) as coend. Moreover if ( E , ψ ) , ( E , ψ ) ∈ J B, C then we have maps pr i : ( E ⊕ E , ψ ⊕ ψ ) −→ ( E i , ψ i ) for i = 1 , , where pr i is the projection.In particular any element of H comes from a map Γ E ,ψ −→ H , where ( E , ψ ) ∈ J B, C . Thus allelements of F Γ , C ( B ) are of the form x E ,ψ for some E ∈ C ⊕ , x ∈ Γ E and ψ ∈ E ( B ) provided thatwe prove that α is an isomorphism.Given an A -module N then Hom A ( H , N ) is A -linearly isomorphic to the set of natural trans-formations of sets β E : E ( B ) −→ Hom A (Γ E , N ) . Since C is additive, by 2.15, those transfor-mations are automatically R -linear. Given b ∈ B there is an A -linear map φ b ∈ End A ( H ) asdescribed in the last sentence in the statement. A direct check shows that, given an A -module N , the map Hom A ( φ b , N ) : Hom A ( H , N ) −→ Hom A ( H , N ) sends an R -linear natural transfor-mation β E : E ( B ) −→ Hom A (Γ E , N ) to β E ◦ b id E . This easily implies that φ : B −→ End A ( H ) makes H into a B ⊗ R A -module and that α : H −→ F Γ , C ( B ) is also B -linear. Since B ⊗ R A -linear maps H −→ N , for N ∈ Mod B ⊗ R A , corresponds to B -linear natural transformations E ( B ) −→ Hom A (Γ E , N ) and thus to A -linear natural transformations Γ E −→ Hom B ( E ( B ) , N ) as described above, one can check directly (using 2.3) that Hom B ⊗ R A ( α, N ) induces the identityon Hom L R ( C ,A ) (Γ − , Hom B ( − ( B ) , N )) . This implies that α : H −→ F Γ , C ( B ) is an isomorphism. (cid:3) Sheafifying R -linear monoidal functors. In this section we show how “ring structures”on a quasi-coherent sheaf over X correspond to “monoidal” structures on the correspondingYoneda functor.We start setting up some definitions: Definition 2.18.
Let C and D be R -linear symmetric monoidal categories. A (contravariant) pseudo-monoidal functor Ω : C −→ D is an R -linear (and contravariant) functor together witha natural transformation ι Ω V,W : Ω V ⊗ Ω W −→ Ω V ⊗ W for V, W ∈ C A (contravariant) pseudo-monoidal functor
Ω : C −→ D is HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 15 symmetric or commutative if for all V, W ∈ C the following diagram is commutative Ω V ⊗ Ω W Ω V ⊗ W Ω W ⊗ Ω V Ω W ⊗ Vι Ω V,W ι Ω W,V where the vertical arrows are the obvious isomorphisms;2) associative if for all
V, W, Z ∈ C the following diagram is commutative Ω V ⊗ Ω W ⊗ Ω Z Ω V ⊗ W ⊗ Ω Z Ω V ⊗ Ω W ⊗ Z Ω V ⊗ W ⊗ Zι Ω V,W ⊗ id ι Ω V,W ⊗ Z id ⊗ ι Ω W,Z ι Ω V ⊗ W,Z If I and J are the unit objects of C and D respectively, a unity for Ω is a morphism J −→ Ω I such that, for all V ∈ C , the compositions Ω V ⊗ J id ⊗ −−−→ Ω V ⊗ Ω I ι Ω V,I −−−→ Ω V ⊗ I −→ Ω V and J ⊗ Ω V ⊗ id −−−→ Ω I ⊗ Ω V ι Ω I,V −−−→ Ω I ⊗ V −→ Ω V coincide with the natural isomorphisms Ω V ⊗ J −→ Ω V and J ⊗ Ω V −→ Ω V respectively. A(contravariant) monoidal functor Ω : C −→ D is a symmetric and associative pseudo-monoidal(contravariant) functor with a unity .A morphism of pseudo-monoidal functors (Ω , ι Ω ) −→ (Γ , ι Γ ) , called a monoidal morphism ortranformation, is a natural transformation Ω −→ Γ which commutes with the monoidal structures ι ∗ . A morphism of monoidal functors is a monoidal transformation preserving the unities. Definition 2.19.
We define the categories: • Rings A X , whose objects are B ∈ QCoh A X with an A -linear map m : B ⊗ O X A B −→ B ,called the multiplication; • QAlg A X , as the (not full) subcategory of Rings A X whose objects are B with a commu-tative, associative multiplication with a unity and the arrows are morphisms preservingunities;We also set Rings X = Rings R X and QAlg X = QAlg R X .Let D be a monoidal subcategory of QCoh X , that is a subcategory such that O X ∈ D andfor all E , E ′ ∈ D we have E ⊗ E ′ ∈ D . We define the category PML R ( D , A ) (resp. ML R ( D , A ) ),whose objects are Γ ∈ L R ( D , A ) with a pseudo-monoidal (resp. monoidal) structure. Remark . If q : X A −→ X is the projection, the equivalence QCoh X A −→ QCoh A X extendsto an equivalence q ∗ : QRings X A −→ QRings A X Indeed, if
G ∈
QCoh X A then q ∗ ( G ⊗ O X A G ) = q ∗ G ⊗ O X A q ∗ G (see 1.11). We will use the followingnotation, which is somehow implicit in the definition of QAlg A X : a sheaf B ∈ QRings A X with B ≃ q ∗ B ′ is associative (resp. commutative, has a unity, ...) if B ′ has the same property.If B ∈ QRings A X with multiplication m , then the composition B ⊗ O X B −→ B ⊗ O X A B m −−→ B induces a ring structure on B as an O X -module, i.e. B ∈ QRings X . Moreover B ∈ QRings A X is associative (resp. commutative, has a unity) if and only if B has the sameproperty. If B ∈ QAlg A X we can form the relative spectrum Spec B over X A and also over X .The final result is the same. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 16
Remark . For B ∈ Rings A X or Γ ∈ PMon R ( D , A ) having a unity is a property, not anadditional datum. Indeed in both cases unities are unique.Let D be a monoidal subcategory of QCoh X . If B ∈ Rings A X with multiplication m , weendow Ω B ∈ L R ( D , A ) with the pseudo monoidal structure ι B : Hom ( E , B ) ⊗ A Hom ( E , B ) −→ Hom ( E⊗E , B ⊗ O X A B ) Hom ( E⊗E ,m ) −−−−−−−−−→ Hom ( E⊗E , B ) , E , E ∈ D that is ι B E , E ( φ ⊗ ψ ) = m ◦ ( φ ⊗ ψ ) . If ∈ B is a unity then we set B = 1 ∈ Ω B O X = H ( B ) Proposition 2.22.
The structures defined above yield an extension of the functor Ω ∗ : QCoh A X −→ L R ( D , A ) to a functor Ω ∗ : QRings A X −→
PML R ( D , A ) . Moreover if B ∈ Rings A X is associa-tive (resp. commutative, has a unity ∈ B ) then Ω B is associative (resp. symmetric, has unity B ∈ Ω B O X ). In particular we also get a functor Ω ∗ : QAlg A X −→ ML R ( D , A ) .Proof. The proof is elementary and it consists of applying the definitions. (cid:3)
Let C be a small monoidal subcategory of QCoh X . Given Γ ∈ PMon R ( C , A ) with monoidalstructure ι , we define the multiplication m Γ : A Γ , C ⊗ O X A A Γ , C −→ A Γ , C on A Γ , C = F Γ , C by A Γ , C ( B ) ⊗ B ⊗ R A A Γ , C ( B ) ∋ x E ,ψ ⊗ x E ,ψ −→ ι E , E ( x ⊗ x ) E⊗E ,ψ ⊗ ψ ∈ A Γ , C ( B ) where Spec B −→ X is a map, E , E ∈ C , ψ ∈ E ( B ) , ψ ∈ E ( B ) , x ∈ Γ E , x ∈ Γ E (see 2.5). Wecontinue to denote by A Γ , C the sheaf F Γ , C together with the multiplication map just defined.If ∈ Γ O X is a unity we set Γ ∈ A Γ , C the image of under the morphism Γ O X −→ Ω A Γ , C O X = H ( A Γ , C ) . Proposition 2.23.
The structures defined above yield an extension of the functor F ∗ , C : L R ( C , A ) −→ QCoh A X to a functor A ∗ , C : PML R ( C , A ) −→ Rings A X which is left adjoint to Ω ∗ : Rings A X −→
PML R ( C , A ) . More precisely, if B ∈ Rings A X then the morphism δ B : A Ω B , C −→ B pre-serves multiplications and unities, while if Γ ∈ PMon R ( C , A ) then the natural transformation γ Γ : Γ −→ Ω A Γ , C is monoidal and preserves unities (see 2.5).If Γ ∈ PMon R ( C , A ) is associative (resp. symmetric, has a unity ∈ Γ O X ) then A Γ , C is asso-ciative (resp. commutative, has unity Γ ∈ A Γ , C ). In particular we get a functor A ∗ , C : ML R ( C , A ) −→ QAlg A X which is left adjoint to Ω ∗ : QAlg A X −→ ML R ( C , A ) .Proof. A lot of properties have to be checked. All of them are very easy, because they consistonly in the application of the definitions. We therefore leave them to the reader. (cid:3) Yoneda embeddings.
In this section we address the problem of when the Yoneda functor QCoh A X −→ L R ( C , A ) is fully faithful and describe its essential image. This will led us to the notion of generatingcategory and left exactness for functors in L R ( C , A ) .We fix an R -algebra A and a fibered category X . We assume that our fibered category X ispseudo-algebraic, that is we assume there exists a representable map X −→ X from a schemewhich is an fpqc covering. We also fix a small subcategory C of QCoh X . Definition 3.1.
Let
D ⊆
QCoh X be a subcategory. A sheaf G ∈
QCoh X is generated by D ifthere exists a surjective morphism M i ∈ I E i −→ G where I is a set and E i ∈ D for all i ∈ I . A sheaf G ∈
QCoh A X is generated by D if it is so asan object of QCoh X . Equivalently, a sheaf G ∈
QCoh X A is generated by D if h ∗ G ∈
QCoh X isso, where h : X A −→ X is the projection. We define QCoh D A X as the subcategory of QCoh A X HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 17 of sheaves G generated by D and such that, for all maps E ψ −−→ G with E ∈ D ⊕ , also Ker ψ isgenerated by D .If D ′ is another subcategory of QCoh X we will say that D generates D ′ if all quasi-coherentsheaves in D ′ are generated by D . Remark . Consider a set of morphisms { U j = Spec B j −→ X } j ∈ J such that ⊔ j U j −→ X isan atlas. By 1.4 we have the following characterizations. If G ∈
QCoh X then G is generated by D if and only if ∀ j ∈ J, x ∈ G ( B j ) , ∃E ∈ D ⊕ , φ ∈ E ( B j ) , E u −−→ G such that u ( φ ) = x If E ∈ D ⊕ and ψ : E −→ G is a map then Ker ψ is generated by D if and only ∀ j ∈ J, y ∈ E ( B j ) with ψ ( y ) = 0 , ∃E ∈ D ⊕ , φ ∈ E ( B j ) , E v −−→ E such that ψv = 0 , v ( φ ) = y In particular if
G ∈
QCoh D X , H ∈
QCoh X is generated by D and H α −−→ G is a map then Ker α is generated by D . Proposition 3.3. If D is a subcategory of QCoh X then the category QCoh D A X is stable by directsums. In particular D ⊆
QCoh D X ⇐⇒ D ⊕ ⊆ QCoh D X .Proof. Let F , G ∈
QCoh D A X . Clearly F ⊕G is generated by D . Now consider a map α : E −→ F ⊕G with
E ∈ D ⊕ and write α = φ ⊕ ψ . By 3.2 it follows that Ker α = Ker ( φ | Ker ( ψ ) : Ker ψ −→ F ) is generated by D because Ker ( ψ ) is generated by D and F ∈
QCoh D A X . (cid:3) If g : U = Spec B −→ X is a map from a scheme then ( J B, C ) op (see 2.17) is not a filteredcategory in general. Thus if Γ ∈ L R ( C , A ) and x E ,φ ∈ F Γ , C ( B ) it is very difficult to understandwhen x E ,φ is zero in F Γ , C ( B ) . Luckily, under some hypothesis this is possible. Lemma 3.4.
Assume
C ⊆
QCoh C X . Then for all flat and representable maps g : Spec B −→ X the category ( J B, C ⊕ ) op is filtered. In this case, given Γ ∈ L R ( C , A ) , an element x E ,φ ∈ F Γ , C ( B ) for ( E , φ ) ∈ J B, C is zero if and only if there exists ( E , φ ) u −−→ ( E , φ ) in J B, C such that Γ u ( x ) = 0 .Proof. By 2.14 we can assume C = C ⊕ . By 2.17 we have to prove that for all maps α, β : ( E , φ ) −→ ( E , φ ) in J B, C there exists u : ( E ′ , φ ′ ) −→ ( E , φ ) in J B, C such that αu = βu . Let K = Ker (( α − β ) : E −→ E ) , so that φ ∈ K ( B ) since g is flat. By assumption K is generated by C and, since C is additive, there exist E ′ ∈ C , a map u : E ′ −→ K and φ ′ ∈ E ′ ( B ) such that u ( φ ′ ) = φ . So ( E ′ , φ ′ ) u −−→ ( E , φ ) is an arrow in J B, C such that ( α − β ) u = 0 as required. The last claim followsfrom 2.17 and the fact that J op B, C is filtered. (cid:3) In what follows we work out sufficient (and sometimes necessary) conditions for the surjectivityor injectivity of δ G : F Ω G , C −→ G . Recall that δ G ( u E ,φ ) = u ( φ ) for E ∈ C ⊕ , Spec B −→ X , φ ∈ E ( B ) , u ∈ Ω GE = Hom X ( E , G ) (see 2.6). Lemma 3.5. If G ∈
QCoh A X the map δ G : F Ω G , C −→ G is surjective if and only if G is generatedby C .Proof. By 2.14 we can assume C = C ⊕ . Let { g i : U i = Spec B i −→ X } be a set of maps suchthat ⊔ i U i −→ X is an atlas. By 1.4 δ G is surjective if and only if δ G ,U i : F Ω G , C ( U i ) −→ G ( U i ) issurjective for all i ∈ I . By 2.17 Im δ G ,U i is the set of elements of G ( U i ) of the form δ G ( u E ,φ ) = u ( φ ) for E ∈ C ⊕ , φ ∈ E ( U i ) , u ∈ Ω GE = Hom X ( E , G ) . So the claim follows from 3.2. (cid:3) Lemma 3.6.
Let
G ∈
QCoh A X . If for all maps E φ −−→ G with E ∈ C ⊕ the kernel Ker φ isgenerated by C then the map δ G : F Ω G , C −→ G is injective. The converse holds if C ⊆
QCoh C X . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 18
Proof.
By 2.14 we can assume C = C ⊕ . Let { g i : U i = Spec B i −→ X } be a set of maps suchthat ⊔ i U i −→ X is an atlas. By 1.4 δ G is injective if and only if δ G ,U i : F Ω G , C ( U i ) −→ G ( U i ) isinjective for all i ∈ I . We start proving that δ G is injective if the hypothesis in the statement isfulfilled. So let z ∈ Ker δ G ,U i . By 2.17 there exists E ∈ C , φ ∈ E ( U i ) and u : E −→ G such that z = u E ,φ and δ G ( u E ,φ ) = u ( φ ) = 0 . Set K = Ker u . Since φ ∈ K ( U i ) and, by hypothesis, K isgenerated by C there exist E ∈ C , φ ∈ E ( U i ) and a map v : E −→ K such that v ( φ ) = φ . If wedenote by v also the composition E −→ K −→ E we have u E ,φ = (Ω G v ( u )) E ,φ = ( uv ) E ,φ = 0 E ,φ = 0 in F Γ , C ( U i ) Assume now that δ G is injective and C ⊆
QCoh C X and let u : E −→ G be a map with
E ∈ C . Wehave to prove that K = Ker u is generated by C . If φ ∈ K ( U i ) ⊆ E ( U i ) , then u ( φ ) = δ G ( u E ,φ ) = 0 .So u E ,φ = 0 and the conclusion follows from 3.2 and 3.4. (cid:3) In general we can still conclude that:
Proposition 3.7. If E ∈ C ⊕ then the map δ E : F Ω E , C −→ E is an isomorphism.Proof. By 2.14 we can assume C = C ⊕ . Let H ∈
QCoh X . The mapHom X ( E , H ) Ω ∗ −−→ Hom L R ( C ,R ) (Ω E , Ω H ) ≃ Hom X ( F Ω E , C , H ) maps id E to δ E and thus is induced by δ E . By the enriched Yoneda’s lemma or a direct check wesee that the above map and therefore δ E are isomorphisms. (cid:3) Theorem 3.8.
Let D A be the subcategory of QCoh A X of sheaves G such that δ G : F Ω G , C −→ G is an isomorphism. Then D A is an additive category containing QCoh C A X , C ⊕ ⊆ D R and thefunctor Ω ∗ : D A −→ L R ( C , A ) is fully faithful. Moreover if C ⊆
QCoh C X then D A = QCoh C A X .Proof. The category D A is additive because Ω ∗ and F ∗ , C are additive. All the other claimsfollows from 3.5, 3.6, 3.7 and the fact that δ G is the counit of an adjiunction. (cid:3) Now we want to address the problem of what is the essential image of the Yoneda functor Ω ∗ : QCoh A X −→ L R ( C , A ) . We will see that if F ∈
QCoh A X the associated Yoneda functor Ω F is always “left exact” and we will give sufficient conditions assuring that “left exact” functorsin L R ( C , A ) are Yoneda functors associated with some quasi-coherent sheaf on X . Since C is notabelian, we introduce an ad hoc notion of left exactness. Definition 3.9.
Let D be a subcategory of QCoh X . A test sequence for D is an exact sequence(3.1) M k ∈ K E k −→ M j ∈ J E j −→ E −→ with E , E j , E k ∈ D for all j ∈ J, k ∈ K in QCoh X given by maps u j : E j −→ E , u kj : E k −→ E j such that for all k ∈ K the set { j ∈ J | u kj = 0 } is finite. We will also say that it is a test sequence for E ∈ D . A finite test sequencefor
E ∈ D is an exact sequence E ′′ −→ E ′ −→ E −→ with E ′ , E ′′ ∈ D ⊕ Given Γ ∈ L R ( D , A ) we say that Γ is exact on the test sequence (3.1) if the sequence(3.2) ( x j ) j ( P j Γ u kj ( x j )) k E Y j ∈ J Γ E j Y k ∈ K Γ E k x (Γ u j ( x )) j HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 19 is exact. We say that Γ is left exact if it is left exact on all short exact sequences in D . We defineLex R ( D , A ) as the subcategory of L R ( D , A ) of functors exact on all test sequences in D . Remark . Notice that the sequence (3.2) is a complex because Γ is R -linear. Moreover weshould warn the reader that the sequence (3.2) is not obtained applying Γ on the test sequence,unless J and K are finite, even if Γ is defined (or extended) to the whole QCoh X . The problemis that Γ does not necessarily transforms infinite direct sums in products. Proposition 3.11.
Let
D ⊆
QCoh X be a subcategory. If F ∈
QCoh A X then Ω F ∈ Lex R ( D , A ) .Proof. It is enough to apply Hom X ( − , F ) to the test sequence (3.1) and observe thatHom X ( M i E i , F ) ≃ Y i Hom ( E i , F ) ≃ Y i Ω FE i (cid:3) Proposition 3.12.
The functor Ω ∗ : QCoh A X −→ L R ( C , A ) is left exact. If C ⊆
QCoh C X then F ∗ , C : L R ( C , A ) −→ QCoh A X is exact.Proof. For the first claim it is enough to use that Hom X ( E , − ) is left exact. For the last partof the statement consider a set of maps { U i = Spec B i g i −−→ X } i ∈ I such that ⊔ i U i −→ X isan atlas. Let also Γ ′ −→ Γ −→ Γ ′′ be an exact sequence in L R ( C , A ) . By 2.17 the sequence F Γ ′ , C ( B i ) −→ F Γ , C ( B i ) −→ F Γ ′′ , C ( B i ) are exact for all i ∈ I because limit of exact sequences Γ ′E ,φ −→ Γ E ,φ −→ Γ ′′E ,φ over the category ( J B i , C ) op , which is filtered thanks to 3.4. Applying 1.4we get the result. (cid:3) Recall that if Γ ∈ L R ( C , A ) then γ Γ , E : Γ E −→ Ω F Γ , C E = Hom X ( E , F Γ , C ) is given by γ Γ , E ( x )( φ ) = x E ,φ for E ∈ C , x ∈ Γ E , Spec B −→ X and φ ∈ E ( B ) (see 2.6). Lemma 3.13.
Assume
C ⊆
QCoh C X . If Γ ∈ Lex R ( C , A ) and the map Ω ∗ ◦ F ∗ , C ( γ Γ ) : Ω F Γ , C −→ Ω F Ω F Γ , C , C is an isomorphism then the natural transformation γ Γ : Γ −→ Ω F Γ , C is an isomorphism.Proof. Let { U i = Spec B i g i −−→ X } i ∈ I be a set of maps such that ⊔ i U i −→ X is an atlas andlet Ψ ∈ L R ( C , A ) and x ∈ Ker γ Ψ , E for some E ∈ C . We are going to prove that there exists asurjective map µ = ⊕ j µ j : ⊕ j ∈ J E j −→ E with E j , E ∈ C such that Ψ u j ( x ) = 0 for all j .If φ ∈ E ( U i ) , by 3.4 and the fact that γ Ψ , E ( x )( φ ) = x E ,φ is zero in F Ψ , C ( U i ) , there exists ( E φ , y φ ) ∈ J B i , C and a map ( E φ , y φ ) u φ −−→ ( E , φ ) such that Ψ u φ ( x ) = 0 . Consider the induced map M i ∈ I M φ ∈E ( U i ) E φ −→ E which is surjective by 1.4. Writing all the E φ ∈ C ⊕ as sums of sheaves in C we get the desiredsurjective map.We return now to the proof of the statement. Given x ∈ Ker γ Γ , E and considering a surjection µ as above for Ψ = Γ , we can conclude that x = 0 by 3.24. This means that the naturaltransformation γ Γ : Γ −→ Ω F Γ , C is injective. Set now Π =
Coker γ Γ , C . By 3.12 we have an exactsequence −→ F Γ , C F ∗ , C ( γ Γ ) −−−−−−→ F Ω F Γ , C −→ F Π , C −→ This is a split sequence because the composition of F ∗ , C ( γ Γ ) : F Γ , C −→ F Ω F Γ , C , C and δ F Γ , C : F Ω F Γ , C , C −→F Γ , C is the identity. So Ω ∗ maintains the exactness of the above sequence and therefore Ω F Π , C = Coker (Ω ∗ ◦ F ∗ , C ( γ Γ )) = 0 We want to prove that
Π = 0 . Let x ∈ Π E for E ∈ C . Since Ω F Π , C = 0 we have x ∈ Ker γ Π , E .Consider a surjection µ = ⊕ j µ j constructed as above starting from x ∈ Π E and Ψ = Π . By
HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 20 µ can be extended to a test sequence ⊕ k E k −→ ⊕ j E i µ −−→ E because C ⊆
QCoh C X . Since Ω F Γ , C ∈ Lex R ( C , A ) by 3.11 we get a commutative diagram E Ω F Γ , C E Π E Y j ∈ J Γ E j Y j ∈ J Ω F Γ , C E j Y j ∈ J Π E j Y k ∈ K Γ E k Y k ∈ K Ω F Γ , C E k β in which all the rows and the first two columns are exact. By diagram chasing it is easy toconclude that β is injective. Since by construction β ( x ) = 0 we can conclude that x = 0 . (cid:3) Definition 3.14.
We define Lex C R ( A ) as the subcategory of Lex R ( C , A ) of functors Γ such that F Γ , C ∈ QCoh C A X . Theorem 3.15.
Assume
C ⊆
QCoh C X . Then the functors Ω ∗ : QCoh C A X −→
Lex C R ( A ) and F ∗ , C : Lex C R ( A ) −→ QCoh C A X are quasi-inverses of each other.Proof. Let Γ ∈ L R ( C , A ) be such that F Γ , C ∈ QCoh C A X . The composition F Γ , C F ∗ , C ( γ Γ ) −−−−−−→ F Ω F Γ , C , C δ F Γ , C −−−−→ F Γ , C is the identity and δ F Γ , C is an isomorphism since F Γ , C ∈ QCoh C A X by 3.8. Thus F ∗ , C ( γ Γ ) andtherefore Ω ∗ ◦ F ∗ , C ( γ Γ ) are isomorphisms. By 3.13 we can conclude that if Γ ∈ Lex C R ( A ) then γ Γ : Γ −→ Ω F Γ , C is an isomorphism. The result then follows from 3.8 and 3.11. (cid:3) The following result allow us to extend results from small subcategories of QCoh X to anysubcategory. Proposition 3.16.
The category
QCoh X is generated by a small subcategory. Equivalently QCoh X has a generator, that is there exists E ∈
QCoh X such that {E} generates QCoh X .Proof. Follows from 1.4 and [SP014, Tag 0780]. (cid:3)
Remark . If D ⊆
QCoh X generates QCoh X there always exists a small subcategory D ⊆ D that generates QCoh X . Indeed if E is a generator of QCoh X it is enough to take a subset ofsheaves in D that generates E . Theorem 3.18.
Let
D ⊆
QCoh X be a subcategory that generates QCoh X . Then the functor Ω ∗ : QCoh A X −→
Lex R ( D , A ) is an equivalence of categories and, if D is small, F ∗ , D : Lex R ( D , A ) −→ QCoh A X is a quasi-inverse. In particular if D ⊆ D is a subcategory that generates
QCoh X the restriction functor Lex R ( D , A ) −→ Lex R ( D , A ) is an equivalence. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 21
Proof. If D is small we have QCoh A X = QCoh D A X , Lex R ( D , A ) = Lex D R ( A ) and everythingfollows from 3.15. In particular the restriction Lex R ( D , A ) −→ Lex R ( D , A ) is an equivalenceif D ⊆ D , they are small and they generate QCoh X . Assume now that D is not neces-sarily small and consider a small subcategory C ⊆ D that generates QCoh X , which existsthanks to 3.17. The proof of the statement is complete if we prove that the restriction functorLex R ( D , A ) −→ Lex R ( C , A ) is an equivalence. Given a set I ⊆ D we set C I = C ∪ I ⊆ D . Forall sets I , quasi-coherent sheaves are generated by C I . Note that we have the restriction func-tor − | C I : Lex R ( D , A ) −→ Lex R ( C I , A ) and the composition QCoh A X Ω ∗ −−→ Lex R ( D , A ) − |C I −−−→ Lex R ( C I , A ) is an equivalence for all I . In particular − | C : Lex R ( D , A ) −→ Lex R ( C , A ) is essen-tially surjective. We will conclude by proving that it is fully faithful. Let Γ , Γ ′ ∈ Lex R ( D , A ) .If I ⊆ I the restriction functor Lex R ( C I , A ) −→ Lex R ( C I , A ) is an equivalence of categories. Inparticular the map Hom (Γ |C I , Γ ′|C I ) −→ Hom (Γ |C I , Γ ′|C I ) is bijective. Using this, it is elementaryto prove that also the map Hom (Γ , Γ ′ ) −→ Hom (Γ |C , Γ ′|C ) is bijective. (cid:3) As a corollary we recover Gabriel-Popescu’s theorem for the category QCoh X . Theorem 3.19. [Gabriel-Popescu’s theorem] If E is a generator of QCoh X then the functor Hom X ( E , − ) : QCoh
X −→
Mod right ( End X ( E )) is fully faithful and has an exact left adjoint.Proof. It follows from 3.18 and 3.12 with D = C = {E} . (cid:3) Before showing other applications of 3.18 we want to present a cohomological interpretationof the functors in Lex R ( C , A ) , which will allow us to show that it is often enough to consider justfinite test sequences instead of arbitrary test sequences. Remark . In an abelian category A , given X, Y ∈ A we can always define the abelian groupExt ( X, Y ) as the group of extensions (regardless if A has enough injectives) and it has the usualnice properties on short exact sequences. In order to avoid set-theoretic problems one shouldrequire that A is locally small and that, given X, Y ∈ A , Ext ( X, Y ) is a set. This is the casefor A = L R ( C , R ) , for instance by looking at the cardinalities of the Γ E for Γ ∈ L R ( C , R ) and E ∈ C . Definition 3.21.
Given a surjective map µ = ⊕ j µ j : ⊕ j E j −→ E with E , E j ∈ C we set Ω µ = ⊕ j Ω µ j : ⊕ j Ω E j −→ Ω E . A functor Γ ∈ L R ( C , A ) is cohomologically left exact on µ if(3.3) Hom L R ( C ,R ) (Ω E / Im (Ω µ ) , Γ) =
Ext L R ( C ,R ) (Ω E / Im (Ω µ ) , Γ) = 0
It is cohomologically left exact if it is so on all surjections µ as above.We setup some notation. We denote by Φ C ( E ) for E ∈ C the set of subfunctor of Ω E of theform Im (Ω µ ) for some surjective map µ : ⊕ j E i −→ E with E j ∈ C and by Φ C the disjoint union ofall the Φ C ( E ) . Given ∆ = Im (Ω µ ) ∈ Φ C and Γ ∈ L R ( C , A ) we will say that Γ is cohomologicallyleft exact on ∆ if it is cohomologically left exact on µ . Notice that, using Yoneda’s lemma, wehave an A -linear isomorphismHom L R ( C ,R ) ( ⊕ j Ω E j , Γ) ≃ Y j Γ E j for all E j ∈ C , Γ ∈ L R ( C , A ) In particular Hom L R ( C ,R ) (Ω E , − ) , which is the evaluation in E , is exact, which implies thatExt L R ( C ,R ) (Ω E , − ) = 0 . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 22
A map ⊕ j E j −→ ⊕ k E k is called locally finite if, for all j , the restriction E j −→ ⊕ k E k factorsthrough a finite sub-sum. Composition of locally finite maps is locally finite and, using Yoneda’slemma, we obtain a functorial mapHom L R ( C ,R ) ( ⊕ j Ω E j , ⊕ k Ω E k ) −→ Hom ( ⊕ j E j , ⊕ k E k ) which is an isomorphism onto the set of locally finite maps.Let µ : ⊕ j E j −→ E be a surjective map and set ∆ = Im (Ω µ ) . Notice that ∆ E ′ is the setof maps E ′ −→ E which factors through µ via a locally finite map E ′ −→ ⊕ j E j . Given a map u : E −→ E in C we set u − (∆) = ∆ × Ω E Ω E ⊆ Ω E : u − (∆) E ′ is the set of maps E ′ −→ E suchthat E ′ −→ E −→ E factors through µ via a locally finite map E ′ −→ ⊕ j E j . Notice that if C ⊆
QCoh C X then u − (∆) ∈ Φ C ( E ) . Indeed, if H is the kernel of µ ⊕ ( − u ) : ( ⊕ j E j ) ⊕ E −→ E then all elements of H on some object Spec B −→ X lies in a finite subsum of ( ⊕ j E j ) ⊕ E . By3.2 H is the image of a locally finite map ⊕ q E q −→ ( ⊕ j E j ) ⊕ E and, since H −→ E is surjective,the induced map µ ′ : ⊕ E ′ ⊕ φ ∈ u − (∆) E′ E ′ −→ E is surjective and clearly u − (∆) = Im (Ω µ ′ ) . Lemma 3.22.
Let µ : ⊕ j E i −→ E be a surjective map with E j , E ∈ C and Γ ∈ L R ( C , A ) . Thenthere is an exact sequence of A -modules −→ Hom L R ( C ,R ) (Ω E / Im (Ω µ ) , Γ) −→ Γ E −→ Y j Γ E j If T : ⊕ k E k −→ ⊕ j E j µ −−→ E −→ is a test sequence and Γ is exact on T then Γ is cohomologicallyleft exact on µ . The converse holds if the map Hom L R ( C ,R ) ( Ker (Ω µ ) , Γ) −→ Y k Γ E k obtained applying Hom L R ( C ,R ) ( − , Γ) to the map ⊕ k Ω E k −→ Ker (Ω µ ) is injective.Proof. Set ∆ = Im (Ω µ ) , K = Ker (Ω µ ) . Consider the diagramHom (Ω E / ∆ , Γ) Γ E Hom (∆ , Γ) Q j Γ E j Hom ( K, Γ) Ext (Ω E / ∆ , Γ) Ext (Ω E , Γ) Q k Γ E k α βλ The convention here is that β and λ are defined only when a test sequence T as in the statementexists and we will not use them for the first statement. All the other maps are obtained splitting Ω E j −→ Ω E −→ into two exact sequences and applying Hom ( − , Γ) , so that the first line andthe central column are exact. The map α obtained as composition is the map defined in the firstsequence in the statement. In particular the first claim follows. So let’s focus on the second one.The map λ is the second map in the statement while the map β together with α are the mapsdefining the sequence (3.2). Since Ext (Ω E , Γ) = 0 also the second claim follows. (cid:3)
Lemma 3.23.
Let Γ , K ∈ L R ( C , R ) and u : ⊕ q Ω E q −→ K be a map, where E q ∈ C . If for all Ω E −→ K with E ∈ C there exists a surjective map v : ⊕ t E t −→ E with E t ∈ C such that thecomposition ⊕ t Ω E t −→ Ω E −→ K factors thorugh u and Γ is cohomologically left exact on v thenthe map Hom L R ( C ,R ) ( K, Γ) −→ Hom L R ( C ,R ) ( ⊕ q Ω E q , Γ) ≃ Y q Γ E q is injective. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 23
Proof.
Let
E ∈ C and x ∈ K E , which corresponds to a map Ω E −→ K . Consider the data givenby hypothesis with respect to this last map. We have commutative diagrams ⊕ t Ω E t Ω E Hom ( K, Γ) Γ E ⊕ q Ω E q K Q q Γ E q Q t Γ E t Ω v xu δλ γ where the second diagram is obtained by applying Hom ( − , Γ) to the first one. The map λ isthe map in the statement, while γ is the evalutation in x ∈ K E . Thanks to 3.22 and since Γ iscohomologically left exact on v the map δ is injective. So if φ ∈ Hom ( K, Γ) is such that λ ( φ ) = 0 it follows that γ ( φ ) = φ E ( x ) = 0 , as required. (cid:3) Theorem 3.24. If C ⊆
QCoh C X then Lex R ( C , A ) coincides with the subcategory of L R ( C , A ) ofcohomologically left exact functors.Proof. Let µ = ⊕ j µ j : ⊕ j ∈ J E j −→ E be a surjective map with E , E j ∈ C and set ∆ = Im (Ω µ ) , K = Ker (Ω µ ) . By 3.2 there exists a test sequence ⊕ k E k −→ ⊕ j E j µ −−→ E −→ . Using 3.22 wehave to prove that if Γ ∈ L R ( C , R ) is cohomologically left exact then λ : Hom ( K, Γ) −→ Q k Γ E k is injective. We are going to apply 3.23 with respect to the map ⊕ k Ω E k −→ K . If E ∈ C , amap Ω E −→ K is a locally finite map E −→ ⊕ j E j which is zero composed by µ , or, equivalently,mapping in the image of ⊕ k E k −→ ⊕ j E j . Consider the kernel H of the difference of the maps ⊕ k E k −→ ⊕ j E j and E −→ ⊕ j E j . Since this difference map is locally finite, C ⊆
QCoh C X andusing 3.2 there is a surjective map ⊕ t E t −→ H with E t ∈ C such that ⊕ t E t −→ ⊕ k E k is locallyfinite and ⊕ t E t −→ E is surjective. This gives the desired factorization for applying 3.23. (cid:3) We now show how to reduce the number of test sequences in order to check when a Γ ∈ L R ( C , A ) belongs to Lex R ( C , A ) . The following is the key lemma: Lemma 3.25.
Let Φ ′ ⊆ Φ C such that, for all E ∈ C and ∆ ∈ Φ C ( E ) there exists ∆ ′ ∈ Φ ′ ∩ Φ C ( E ) such that ∆ ′ ⊆ ∆ (inside Ω E ). If Γ ∈ L R ( C , A ) , C ⊆
QCoh C X and Γ is cohomologically leftexact on all the elements of Φ ′ then Γ is cohomologically left exact.Proof. Consider ∆ ∈ Φ C ( E ) , ∆ ′ ⊆ ∆ with ∆ ′ ∈ Φ ′ and the exact sequence −→ ∆ / ∆ ′ −→ Ω E / ∆ ′ −→ Ω E / ∆ −→ . Applying Hom ( − , Γ) and using that Ext (Ω E , Γ) = 0 , the only nontrivial vanishing to check is Hom (∆ / ∆ ′ , Γ) = 0 . Write ∆ ′ = Im (Ω u ) , where u : ⊕ q E q −→ E .We can apply 3.23 to K = ∆ and the map Ω u : ⊕ q Ω E q −→ ∆ : if Ω E ′ −→ ∆ ⊆ Ω E is amap corresponding to ψ : E ′ −→ E , then ψ − (∆ ′ ) ∈ Φ C and, by hypothesis, we can find Φ ′ ∋ Im (Ω v ) ⊆ ψ − (∆ ′ ) ; the last inclusion tells us that v is the factorization required for 3.23. Thusthe map Hom (Ω u , Γ) :
Hom (∆ , Γ) −→ Hom ( ⊕ q Ω E q , Γ) is injective and, since ∆ ′ = Im (Ω u ) ⊆ ∆ ,this is also true for the restriction Hom (∆ , Γ) −→ Hom (∆ ′ , Γ) , whose kernel is Hom (∆ / ∆ ′ , Γ) ,as required. (cid:3) Proposition 3.26.
Let
D ⊆
QCoh X be a subcategory and Γ ∈ L R ( D , A ) . If Γ ∈ Lex R ( D , A ) then Γ is exact on finite test sequences in D and transforms arbitrary direct sums in D intoproducts. The converse holds if one of the following conditions is satisfied: • the category D is stable by arbitrary direct sums; • all the sheaves in D are finitely presented, D ⊆
QCoh D X and X is quasi-compact. Inthis case Γ ∈ Lex R ( D , A ) if and only if it is cohomologically left exact on all surjectivemaps E ′ −→ E with E ∈ D and E ′ ∈ D ⊕ . HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 24
In any of the above cases, if moreover D is additive and all surjections in D have kernel in D then Lex R ( D , A ) is the subcategory of L R ( D , A ) of left exact functors which transforms arbitrarydirect sums in products.Proof. If Γ ∈ Lex R ( D , A ) then it is clearly exact on finite test sequences. Given a set {E j ∈ D} j ∈ J set E = L j E j . If E ∈ D , then the sequence −→ M j ∈ J E j id E −−→ E −→ is a test sequence and therefore we get that the natural map Γ E −→ Q j Γ E j is an isomorphism.If D is stable by arbitrary direct sums is easy to see that the converse holds. Moreover the lastpart of the statement follows easily from the first part.So we focus on the second point and we assume that all the sheaves in D are finitely presented, D ⊆
QCoh D X and that X is quasi-compact. Since the class of finitely presented quasi-coherentsheaves on X modulo isomorphism is a set, we can assume D = C small. Let Φ ′ ⊆ Φ C be thesubset of functors of the form Im (Ω µ ) for some surjective map µ : E ′ −→ E with E ∈ C and E ′ ∈ C ⊕ . The set Φ ′ satisfies the hypothesis of 3.25: if v : ⊕ j ∈ J E j −→ E is a surjective map thenthere exists a finite subset J ⊆ J such that v |E ′ : E ′ = ⊕ j ∈ J E j −→ E is surjective because E isof finite type and X is quasi-compact. In particular, taking into account 3.24, the last claim ofthe second point follows. It remains to show that if Γ ∈ L R ( C , A ) is exact on finite test sequencesthen Γ is cohomologically left exact on all the elements of Φ ′ . Let µ : E ′ −→ E be a surjectivemap with E ∈ C and E ′ ∈ C ⊕ . Since E is finitely presented and E ′ is of finite type it follows thatKer ( µ ) is of finite type and, since X is quasi-compact and C ⊆
QCoh C X , there exists E ′′ ∈ C ⊕ and a surjective map E ′′ −→ Ker ( µ ) . Thus E ′′ −→ E ′ µ −−→ E −→ is a finite test sequence andby 3.22 it follows that Γ is cohomologically left exact on µ as required. (cid:3) There is another characterization of Lex R ( C , A ) in terms of sheaves on a site. Although wewill not use it in this paper, I think it is worth to point out. We refer to [SP014, Tag 00YW] forgeneral definitions and properties. We start by comparing Lex R ( C , A ) and Lex R ( C ⊕ , A ) . Proposition 3.27. If C ⊆
QCoh C X then the equivalence L R ( C ⊕ , A ) ≃ L R ( C , A ) maps Lex R ( C ⊕ , A ) to Lex R ( C , A ) .Proof. We can assume A = R . Let Γ ∈ L R ( C ⊕ , R ) such that Γ ∈ Lex R ( C , R ) and consider Φ ′ ⊆ Φ C ⊕ the set of subfunctors ∆ ⊆ Ω E that can be written as follows: E = E ⊕· · ·⊕E r and thereare surjective maps µ k : ⊕ q E q,k −→ E k for E k , E q,k ∈ C such that ∆ = Im (Ω µ ) ⊕ · · · ⊕ Im (Ω µ r ) .Since for such ∆ we have Ω E / ∆ ≃ ⊕ i (Ω E i / ∆ i ) it follows that Γ is cohomologically left exact on all the elements of Φ ′ . Taking into account 3.24,in order to conclude that Γ ∈ Lex R ( C , R ) we can show that Φ ′ ⊆ Φ C ⊕ satisfies the hypothesis of3.25. So let ∆ = Im (Ω µ ) ∈ Φ C ⊕ where µ : ⊕ q E q −→ E where E , E q ∈ C ⊕ . If E = E ′ ⊕ · · · ⊕ E ′ r with E ′ i ∈ C and ψ i : E ′ i −→ E are the inclusions then ∆ i = ψ − i (∆) ∈ Φ C ⊕ ( E ′ i ) = Φ C ( E ′ i ) and it iseasy to see that Φ ′ ∋ ∆ ⊕ · · · ⊕ ∆ r ⊆ ∆ as required. (cid:3) Proposition 3.28. If C ⊆
QCoh C X and J is the smallest Grothendieck topology on C ⊕ con-taining Φ C ⊕ then Lex R ( C ⊕ , A ) is the category of sheaves of A -modules on ( C ⊕ , J ) which are R -linear.Proof. We can assume A = R and C = C ⊕ . If ∆ ⊆ Ω E is a sieve and f : E ′ −→ E we set f − (∆) = ∆ × Ω E Ω E ′ ⊆ Ω E ′ . Let e J be the set of sieves ∆ ⊆ Ω E of C such that, for all HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 25 Γ ∈ Lex R ( C , R ) and maps f : E ′ −→ E the mapHom ( Sets ) (Ω E ′ , Γ) −→ Hom ( Sets ) ( f − (∆) , Γ) is bijective. Here Hom ( Sets ) denotes the set of natural transformation of functors with valuesin ( Sets ) . The set e J is a Grothendieck topology on C such that all functors in Lex R ( C , R ) aresheaves. Notice that, by 2.15, if A, B ∈ L R ( C , R ) then Hom ( Sets ) ( A, B ) =
Hom L R ( C ,R ) ( A, B ) .Moreover if ∆ ∈ Φ C and f : E ′ −→ E is a map in C then ∆ ′ = f − (∆) ∈ Φ C and if Γ ∈ L R ( C , R ) then, applying Hom L R ( C ,R ) ( − , Γ) on the exact sequence −→ ∆ ′ −→ Ω E ′ −→ Ω E ′ / ∆ ′ −→ andtaking into account that Ext (Ω E ′ , Γ) = 0 we obtain an exact sequence −→ Hom (Ω E ′ / ∆ ′ , Γ) −→ Hom ( Sets ) (Ω E ′ , Γ) −→ Hom ( Sets ) (∆ ′ , Γ) −→ Ext (Ω E ′ / ∆ ′ , Γ) −→ Thus, if J is the smallest topology on C containing Φ C then Φ C ⊆ J ⊆ e J and everything followseasily from 3.24. (cid:3) We now apply 3.18 and 3.26 in some (more) concrete situations.
Theorem 3.29.
The category
Lex R ( QCoh X , A ) is the category of contravariant, R -linear andleft exact functors Γ :
QCoh
X −→
Mod A which transform arbitrary direct sums in products.Moreover the functor Ω ∗ : QCoh A X −→
Lex R ( QCoh X , A ) is an equivalence of categories.Proof. Follows from 3.18 and 3.26 with D = QCoh X . (cid:3) Theorem 3.30.
Let X be a noetherian algebraic stack. The category Lex R ( Coh X , A ) is thecategory of contravariant, R -linear and left exact functors Coh
X −→
Mod A . Moreover thefunctor Ω ∗ : QCoh A X −→
Lex R ( Coh X , A ) is an equivalence of categories.Proof. Follows from 3.18 and 3.26, taking into account that in our assumptions Coh X is anabelian category that generates QCoh X . (cid:3) Theorem 3.31.
Assume that X is quasi-compact and that Loc X generates QCoh X . Then Lex R ( Loc X , A ) is the category of contravariant, R -linear and left exact functors Loc
X −→
Mod A . Moreover the functor Ω ∗ : QCoh A X −→
Lex R ( Loc X , A ) is an equivalence of categories.Proof. Follows from 3.18 and 3.26, taking into account that all surjections in Loc X have kernelsin Loc X . (cid:3) Theorem 3.32.
Let B be an R -algebra and D ⊆
Mod B be a subcategory that generates Mod B ,that is there exists E , . . . , E r ∈ D with a surjective map L i E i −→ B . Then the functor Ω ∗ : Mod ( A ⊗ R B ) −→ Lex R ( D , A ) is an equivalence of categories. Moreover if D ⊆
Loc B then Lex R ( D , A ) = L R ( D , A ) .Proof. If X = Spec B , then QCoh A X ≃
Mod ( A ⊗ R B ) and the first part follows from 3.18.For the last claim, observe that any Γ :
Loc B −→ Mod A is exact because any short exactsequence in Loc B splits. By 3.31 we can conclude that L R ( Loc
B, A ) =
Lex R ( Loc
B, A ) . If now D ⊆
Loc B and Γ ∈ L R ( D , A ) , we can extend it to Γ ∈ L R ( Loc
B, A ) and therefore Γ = Γ |D ∈ Lex R ( D , A ) . (cid:3) HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 26
We want to extend Theorem 3.18 to functors with monoidal structures.
Definition 3.33. If D is a monoidal subcategory of QCoh X we define PMLex R ( D , A ) (resp.MLex R ( D , A ) ) as the subcategory of PML R ( D , A ) (resp. ML R ( D , A ) ) of functors Γ such that Γ ∈ Lex R ( D , A ) . Theorem 3.34.
Let D be a monoidal subcategory of QCoh X that generates it. Then the functors Ω ∗ : Rings A X −→
PMLex R ( D , A ) and Ω ∗ : QAlg A X −→
MLex R ( D , A ) (see 2.22) are equivalence of categories. If D is small a quasi inverse is given by A ∗ , D : PMLex R ( D , A ) −→ Rings A X and A ∗ , D : MLex R ( D , A ) −→ QAlg A X respectively (see 2.23). Moreover if D ⊆ D is a monoidal subcategory that generates
QCoh X the restriction functors PMLex R ( D , A ) −→ PMLex R ( D , A ) and MLex R ( D , A ) −→ MLex R ( D , A ) are equivalences.Proof. Assume that D is small. Then Ω ∗ : Rings A X −→
PMLex R ( D , A ) and A ∗ , D : PMLex R ( D , A ) −→ Rings A X are quasi-inverses of each other because, by 2.23, we have natural transformationsid γ −−→ Ω ∗ ◦ A ∗ , D and A ∗ , D ◦ Ω ∗ β −−→ id which are isomorphisms thanks to 3.18. In particu-lar if D ⊆ D is a monoidal subcategory that generates QCoh X then the restriction functorPMLex R ( D , A ) −→ PMLex R ( D , A ) is an equivalence. Since γ and β preserve unities by 2.23,the same holds if we replace Rings A X by QAlg A X and PMLex R ( D , A ) by MLex R ( D , A ) . Noticethat there exists a small subcategory C ′ ⊆ D that generates QCoh X thanks to 3.17. If I ⊆ D is aset we set C I for the category containing all tensor products with factors in C ∪ I and O X . We havethat C I is a collection of small monoidal subcategories of D , that generate QCoh X and such that C I ⊆ C I ′ if I ⊆ I ′ . If C = C ∅ we can show that the restrictions PMLex R ( D , A ) −→ PMLex R ( C , A ) and MLex R ( D , A ) −→ MLex R ( C , A ) are equivalences by proceeding as in the proof of 3.18. Allthe other claims in the statement follow easily from this fact. (cid:3) Theorem 3.35.
The theorems 3.29, 3.30, 3.31 continue to hold if we replace
Lex R by PMLex R (resp. MLex R ), QCoh A X by QRings A X (resp. QAlg A X ) and the word “functors” by “pseudo-monoidal functors” (resp. “monoidal functors”). Group schemes and representations.
Let G be a flat and affine group scheme over R . In this section we want to interpret theresults obtained in the case X = B R G , the stack of G -torsors for the fpqc topology, which is aquasi-compact fpqc stack with affine diagonal.If A is an R -algebra, by standard theory we have that QCoh A B R G is the category Mod G A of G -comodules over A . Recall that the regular representation R [ G ] of G is by definition the G -comodule p ∗ O G . By definition it comes equipped with a morphism of R -algebras ε : R [ G ] −→ R induced by the unit section of G . Remark . If M ∈ Mod G A then the composition ( M ⊗ R R [ G ]) G −→ M ⊗ R R [ G ] id M ⊗ ε −−−−→ M is an isomorphism. This follows from [Jan87, 3.4] applied to G = H .We start with a criterion to find a set of generators for QCoh B R G . Proposition 4.2.
If the regular representation R [ G ] is a filtered direct limit of modules B i ∈ Mod G R which are finitely presented as R -modules then { B ∨ i } i ∈ I generates Mod G R .Proof. Set B = R [ G ] and ε i : B i −→ R for the composition B i −→ B ε −−→ R and let M ∈ Mod G R . Since filtered direct limits commute with tensor products and taking invariants, by 4.1we have that the limit of the maps ( ε i ⊗ id M ) | ( B i ⊗ M ) G : ( B j ⊗ M ) G −→ M is an isomorphism. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 27
This means that for any m ∈ M there exists i m ∈ I and an element ψ m ∈ ( B i m ⊗ M ) G suchthat ( ε i ⊗ id M )( ψ m ) = m . The map B i ⊗ M −→ Hom ( B i ∨ , M ) is G -equivariant and thereforewe obtain a δ m ∈ Hom G ( B i ∨ , M ) such that δ m ( ε i ) = m . This implies that the map M m ∈ M δ m : M m ∈ M B ∨ i m −→ M is surjective and therefore that M is generated by { B i ∨ } i ∈ I . (cid:3) Remark . The class G R of flat, affine group schemes G over R such that R [ G ] is a directlimit of modules in Loc ( B R G ) is stable by arbitrary products and projective limits. Moreover byconstruction contains all groups which are flat, finite and finitely presented over R , i.e. R [ G ] ∈ Loc ( B R G ) , and thus all profinite groups. Since any G -comodule is the union of the sub G -comodules which are finitely generated R -modules (see [Jan87, 2.13]), we see that G R containsall flat groups defined over a Dedekind domain or a field, such as GL r , SL r and all diagonalizablegroups. Proposition 4.2 tells us that if G ∈ G R then B R G has the resolution property.Let A be an R -algebra. We denote by Loc G A the subcategory of Mod G A of G -comodulesthat are locally free of finite rank (projective of finite type) as A -modules, so that Loc ( B R G ) ≃ Loc G R . We define QAdd G A (QPMon G A , QMon G A ) as the category of covariant R -linear(pseudo-monoidal, monoidal) functors Loc G R −→ Mod A . We set QRings G A for the categoryof M ∈ Mod G A with a G -equivariant map M ⊗ A M −→ M and QAlg G A for the (not full)subcategory of QRings G A of commutative R -algebras. Definition 4.4.
The group G is called linearly reductive if the functor ( − ) G : Mod G R −→ Mod R is exact. Remark . If G is linearly reductive then any short exact sequence in Loc G R splits. Indeed if M −→ N is surjective then Hom GR ( N, M ) −→ Hom GR ( N, N ) is surjective, yielding a G -equivariantsection N −→ M . Theorem 4.6. If B R G has the resolution property then the functors Mod G A −→ QAdd G A, QRings G A −→ QPMon G A, QAlg G A −→ QMon G A which maps M to the functor ( − ⊗ R M ) G : Loc G R −→ Mod A are well defined, fully faithful andhave essential image the subcategory of functors which are left exact on short exact sequences in Loc G R . In particular they are equivalences if G is a linearly reductive group.Proof. Set C = Loc G R . The functor ( − ) ∨ : Loc G R −→ Loc G R is an equivalence and there-fore we get equivalences QAdd G A ≃ L R ( C , A ) , QPMon G A ≃ PML R ( C , A ) and QMon G A ≃ ML R ( C , A ) . Left exact functors are sent to left exact functors. Under those equivalences Ω M corresponds to ( − ⊗ R M ) G because Hom B R G ( E ∨ , M ) ≃ H ( E ⊗ O B R G M ) ≃ ( E ⊗ R M ) G . Thusthe result follows from 3.12, 3.31, 3.35 and 4.5. (cid:3) Tannaka reconstruction for stacks with the resolution property.
Definition 5.1.
A (contravariant) monoidal functor
Ω : C −→ D between symmetric monoidalcategories is called strong if the maps Ω V ⊗ Ω W −→ Ω V ⊗ W are isomorphisms for all V, W ∈ C and the map J −→ F I is an isomorphism, where I and J arethe unities of C and D respectively. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 28
In this section we want to understand what sheaves of algebras correspond to strong monoidalfunctors in the equivalence of 3.34, in the case where D is a subcategory of locally free sheaves.We will consider only fpqc stacks with quasi-affine diagonal, for instance algebraic stacks withquasi-affine diagonal (see [MBL99, Corollary 10.7]) and quasi-separated schemes. This is becauseresolution property is somehow meaningless for other stacks, see for instance Remark (1) in theintroduction of [Tot04]. Definition 5.2. If X is a fiber category over R , C ⊆
Loc X is a monoidal subcategory and A is an R -algebra we define SMex R ( C , A ) as the subcategory of MLex R ( C , A ) of functors Γ whichare strong monoidal and, for all geometric points Spec k −→ Spec A , Γ ⊗ A k ∈ MLex R ( C , k ) .Given a fiber category Y we denote by Fib X , C ( Y ) the category of covariant , R -linear and strongmonoidal functors Γ :
C −→
Loc Y which are exact on all exact sequences E ′′ −→ E ′ −→ E −→ with E ∈ C and E ′ , E ′′ ∈ C ⊕ . We also define Fib X , C as the fiber category over R whose fiber overan R -algebra A is Fib X , C ( Spec A ) and we call P C the functor(5.1) P C : X −→
Fib X , C , ( Spec A s −−→ X ) ( s ∗ : C −→
Loc A ) We will prove the following:
Theorem 5.3.
Let X be a quasi-compact fpqc stack over R with quasi-affine diagonal, A be an R -algebra and C ⊆
Loc X be a monoidal subcategory with duals that generates QCoh X . Thenthe functors ( Spec A Γ , C −→ X ) (Γ : C −→
Mod A ) X ( A ) SMex R ( C , A )( s : Spec A −→ X ) (( s ∗|C ) ∨ : C −→
Loc A ) are well defined and quasi-inverses of each other. In particular the functor P C : X −→
Fib X , C isan equivalence of stacks. An immediate corollary and generalization of Theorem 5.3 is the following.
Corollary 5.4.
Let X be a quasi-compact fpqc stack over R with quasi-affine diagonal, C ⊆
Loc X be a monoidal subcategory with duals that generates QCoh X and Y be a fibered category over R .Then the functor Hom ( Y , X ) −→ Fib X , C ( Y ) , ( Y f −−→ X ) f ∗|C : C −→
Loc ( Y ) is an equivalence of categories.Proof. The map in the statement is obtained applying Hom ( Y , − ) to the functor P C : X −→
Fib X , C , which is an equivalence by 5.3. (cid:3) One of the key points in the proof of statements above is a characterization of the followingstacks.
Definition 5.5. A pseudo-affine stack is a quasi-compact fpqc stack with quasi-affine diagonalsuch that all quasi-coherent sheaves on it are generated by global sections. A map f : Y ′ −→ Y of fibered categories is called pseudo-affine if for all maps T −→ Y from an affine scheme thefiber product T × Y Y ′ is pseudo-affine. Theorem 5.6.
Intersection of quasi-compact open subschemes (thought of as sheaves) of affineschemes are pseudo-affine. Conversely if U is a pseudo-affine stack then it is (equivalent to) asheaf and it is the intersection of the quasi-compact open subschemes of Spec H ( O U ) containingit. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 29
A quasi-affine scheme is pseudo-affine and, conversely, a pseudo-affine stack which is algebraicis quasi-affine (see [Gro13, Proposition 3.1]). In general a pseudo-affine sheaf is not quasi-affine.An example is the sheaf intersection of all the complement of closed points in Spec k [ x, y ] , where k is a field.The last statement of Theorem 5.3 admits an almost converse. Let X be a fibered categoryover R and C ⊆
Loc X be a full monoidal subcategory. We have a functor G ∗ : C −→
Loc ( Fib X , C ) which maps a E ∈ C to the locally free sheaf G E : Fib op X , C −→ ( Ab ) , G E (Γ ∈ Fib X , C ( A )) = Γ E for all R -algebras A In particular P ∗C G E ≃ E for E ∈ C . Notice that, a priori, Fib X , C is not necessarily fibered ingroupoids and therefore the notion of a locally free sheaf on it is not defined (although one caneasily guess the definition). Part of Theorems 5.3 and 5.7 (below) is to prove that, under suitableconditions on C , this is indeed true. Theorem 5.7.
Let X be a quasi-compact fibered category over R and C ⊆
Loc X be a fullmonoidal subcategory with duals and with Sym n E ∈ C for all n ∈ N if E has local rank notinvertible in R . If f : C −→ N is a function then Fib f X , C , the sub-fibered category of Fib X , C offunctors Γ such that rk Γ E = f ( E ) for all E ∈ C , is a quotient stack of a pseudo-affine sheaf bythe action of a (possibily infinite) products of GL n ; in particular it is a quasi-compact fpqc stackin groupoids with affine diagonal. Moreover the subcategory {G E } E∈C ⊆ Loc ( Fib f X , C ) generates QCoh ( Fib f X , C ) and, in particular, Fib f X , C has the resolution property.Let I be the set of functions f : C −→ N such that there exists a geometric point s : Spec L −→X with f ( E ) = rk s ∗ E . Given f : C −→ N then Fib f X , C = ∅ if and only if f ∈ I and, if I is finite,then Fib X , C = G f ∈I Fib f X , C In particular if I is finite then Fib X , C is a quasi-compact fpqc stack in groupoids with affinediagonal, the subcategory {G E } E∈C ⊆ Loc ( Fib X , C ) generates QCoh ( Fib X , C ) and Fib X , C has theresolution property.Remark . The condition that I is finite in 5.7 is not optimal, but at least it covers the casewhere X admits a surjective (on equivalence classes of geometric points) map X ′ −→ X froman algebraic stack whose connected components are open (e.g. X is a connected or Noetherianalgebraic stack). It is not clear if Fib X , C always has the resolution property.We start with a first characterization of pseudo-affine stacks. Notice that the condition thatall quasi-coherent sheaves on X are generated by global section exactly means that the category {O X } generates QCoh X . Proposition 5.9.
Let X π −−→ Spec R be a quasi-compact fpqc stack with quasi-affine diagonal.Then the following conditions are equivalent:1) the stack X is pseudo-affine;2) the map π ∗ π ∗ F −→ F is surjective for all
F ∈
QCoh X ;3) the stack X is equivalent to a sheaf and there exists a flat monomorphism X −→
Spec B ,where B is a ring.In this case the map p : X −→
Spec H ( O X ) is a flat monomorphism, p ∗ : QCoh
X −→
Mod H ( X ) is fully faithful and p ∗ p ∗ ≃ id . Moreover if X × H ( O X ) k = ∅ for all geometric points Spec k −→ Spec H ( O X ) then p is an isomorphism. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 30
Proof.
2) = ⇒ . Given F ∈
QCoh X , take a surjective map R ( I ) −→ π ∗ F . In this case thecomposition O ( I ) X ≃ π ∗ R ( I ) −→ π ∗ π ∗ F −→ F is surjective.
1) = ⇒ . A sheaf F ∈
QCoh X is generated by global sections and the image of π ∗ π ∗ F −→ F contains all of them.
3) = ⇒ . Denote by p : X −→
Spec B the flat monomorphism. We are going to show that δ F : p ∗ p ∗ F −→ F is an isomorphism for all
F ∈
QCoh X . Arguing as in
2) = ⇒ this willconclude the proof. By hypothesis there exists a representable fpqc covering h : Spec C −→ X and we must prove that h ∗ δ F is an isomorphism. Let f = ph : Spec C −→ Spec B be thecomposition, which is flat by hypothesis, and consider the commutative diagramSpec C X × B C Spec C X Spec B s αp fh Since α is a monomorphism with a section, α and s are inverses of each other. Using the factthat f is flat the map h ∗ δ F is given by p ∗ p ∗ F ( Spec C h −−→ X ) = p ∗ F ( Spec C f −−→ Spec B ) = F ( X × B C −→ X ) s ∗ −−→ F ( Spec C h −−→ X ) and therefore it is an isomorphism.
1) = ⇒ . Set B = H ( O X ) and p : X −→
Spec B the induced map. Notice thatL B ( {O X } , B ) ≃ Mod B and under this isomorphism Ω ∗ : QCoh
X −→ L B ( {O X } , B ) and F ∗ , {O X } : L B ( {O X } , B ) −→ QCoh X correspond to p ∗ : QCoh
X −→
Mod B and p ∗ : Mod B −→ QCoh X respectively. Byhypothesis, 3.12 and 3.18 the map p : X −→
Spec B is flat, p ∗ : QCoh
X −→
Mod B is fullyfaithful and p ∗ p ∗ ≃ id.We want to show that X −→
Spec B is fully faithful or, equivalently, that the diagonal X −→ X × B X is an equivalence. Let V = Spec C −→ X be a representable fpqc covering anddenote by s : V −→ X × B V the graph of h . We have Cartesian diagrams V × X V V × B V V × X V V × B V X X × B X V X × B V q ∆ s q and the vertical arrows are representable fpqc coverings. By descent it follows that ∆ is anequivalence if we prove that s is an equivalence.Let f : V −→ X p −−→ Spec B be the composition. Since p is flat also f is flat and, in particular,H ( O X × B V ) = C . Since X × B V −→ X is affine it follows that {O X × B V } generates QCoh ( X × B V ) . Since s is a section of X × B V −→ V , we see that we can assume that p : X −→
Spec B has asection, that we still denote by s : Spec B −→ X . In this case we have to prove that p or s is anequivalence. Notice that this implies the last claim in the statement. Indeed if X × B k = ∅ for allgeometric points then f : V −→ Spec B is an fpqc covering: it follows that the map X −→
Spec B is an equivalence because it has this property fpqc locally.We first prove that p ∗ : QCoh
X −→
Mod B is an equivalence. It suffices to show that, if M ∈ Mod B , then the map γ M : M −→ p ∗ p ∗ M is an isomorphism. Notice that p ∗ γ M is a sectionof the map ( p ∗ p ∗ ) p ∗ M −→ p ∗ M which is an isomorphism. So p ∗ γ M and γ M = s ∗ p ∗ γ M areisomorphisms. Since p ∗ s ∗ : Mod B −→ Mod B is the identity, we can conclude that s ∗ ≃ p ∗ and HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 31 s ∗ ≃ p ∗ . Consider the Cartesian diagram U Spec C Spec B X g s ht where h is a representable fpqc covering. Since X has quasi-affine diagonal it follows that U isa quasi-affine scheme. MoreoverH ( O U ) ≃ g ∗ t ∗ O B ≃ h ∗ s ∗ O B ≃ h ∗ p ∗ O B ≃ C Thus g : U −→ Spec C and s : Spec B −→ X are open immersions. We prove that g is asurjective, which imply that g and s are isomorphisms. Let Z be the complement of U in Spec C with reduced structure. We haveH ( O Z ) ≃ p ∗ h ∗ O Z ≃ s ∗ h ∗ O Z = t ∗ g ∗ O Z = 0 Thus Z is empty as required. (cid:3) Remark . The assumption on the diagonal in 5.9 is necessary: the stack X = B k E , where E is an elliptic curve over a field k , is not a sheaf but QCoh X ≃
QCoh k . Remark . If f : Y −→ X is a pseudo-affine map of pseudo-algebraic fiber categories then it isquasi-compact with affine diagonal and the map f ∗ f ∗ F −→ F is surjective for all
F ∈
QCoh Y .In particular if D ⊆
QCoh X generates QCoh X then f ∗ D generates QCoh Y . The first claimsfollow by standard argument of descent while the second by considering an atlas of X and usingthe characterization of 5.9.If Y −→
Spec A is a map of fibered categories then Y is pseudo-affine if and only if Y −→
Spec A is pseudo-affine, because if Y is pseudo-affine and g : Y ′ −→ Y is an affine map, then Y ′ is an fpqc sheaf with quasi-affine diagonal and O Y ′ = g ∗ O Y generates QCoh Y ′ .Finally, if f : Y ′ −→ Y is a map of fpqc stacks and Y −→ Y is an fpqc atlas such that Y ′ = Y × Y Y ′ −→ Y is pseudo-affine then f is pseudo-affine. Indeed by standard arguments ofdescent we can assume Y = Spec B and Y = Spec B ′ affine, which also implies that Y ′ is quasi-compact with affine diagonal. Moreover since B −→ B ′ is flat we have H ( O Y ′ ) ≃ H ( O Y ′ ) ⊗ B B ′ and therefore we can assume that H ( O Y ′ ) = B and H ( O Y ′ ) = B ′ . In this case Y ′ −→ Y is flatand fully faithful and, since Y is an fpqc stack, it follows that also Y ′ −→ Y is a flat and fullyfaithful. Remark . Let G be a flat and affine group scheme over R such that B R G has the resolutionproperty. Taking into account 5.11, if U is a pseudo-affine sheaf over R with an action of G then [ U/G ] has the resolution property because the map [ U/G ] −→ B R G is pseudo-affine. The sameconclusion follows for a stack X = [ X/G ] , where X is a scheme, if there exists L ∈
Pic ( X ) whosepullback M to X is very ample relatively to R . Indeed X can be written as [ U/G × G m ] where U is the complement of the zero section of M −→ X : the fact that M is very ample tells us that U is quasi-affine. Moreover B ( G × G m ) has the resolution property. Indeed let N be the canonicalinvertible sheaf on B R G m and F ∈
QCoh ( B R G ) . The action of G m yields a decomposition F = M n ∈ Z F n and all F n = ( F ⊗ N ⊗− n ) G m are sub G × G m -representations. If x ∈ F n there exists E ∈
Loc ( B R G ) and a G -equivariant map E −→ F n with x in its image. Thus the map E⊗ N ⊗ n −→ F n ,where E has the trivial action of G m , is G × G m -equivariant and has x in its image. HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 32
The following property is known for algebraic stacks (see [Gro13, Corollary 5.11] and [Tot04,Proposition 1.3]).
Corollary 5.13.
A quasi-compact fpqc stack X with quasi-affine diagonal and with the resolutionproperty has affine diagonal.Proof. Since X is pseudo-algebraic the category Loc X is essentially small. Thus we can considera set R of representatives of locally free sheaves over X . Given E ∈ R we define the sheafFr ( E ) : ( Sch / X ) op −→ ( Sets ) , Fr ( E )( U f −−→ X ) = G n ∈ N Iso U ( O nU , f ∗ E ) It is easy to see that Fr ( E ) −→ X is an affine fpqc covering. In particular g : Fr = Q E∈R Fr ( E ) −→X is also an affine fpqc covering and Fr is quasi-compact with quasi-affine diagonal. Thus it isenough to show that Fr has affine diagonal. Since g is affine by 5.11 g ∗ Loc X generates QCoh Fr.On the other hand if E ∈
Loc X then by construction Fr is a (finite) disjoint union of opensubstacks over which g ∗ E is free, which implies that g ∗ E is generated by global sections. We canconclude that Fr is a pseudo-affine sheaf and thus has affine diagonal. (cid:3) Proof. (of Theorem 5.3). Since X has affine diagonal by 5.13, all morphisms Spec A −→ X areaffine. Therefore the functor X ( A ) −→ QAlg A X which maps s : Spec A −→ X to s ∗ O A is fullyfaithful. By 3.34 and the fact that Ω s ∗ O A E = Hom ( E , s ∗ O A ) ≃ ( s ∗ E ) ∨ for s ∈ X ( A ) , E ∈ C we can conclude that the functor X ( A ) −→ SMex R ( C , A ) , s ( s ∗|C ) ∨ is well defined and fullyfaithful. Thus everything follows if we prove that, given Γ ∈ SMex R ( C , A ) , the composition p : Spec A Γ , C −→ X A −→ Spec A is an isomorphism. Set A = A Γ , C , Y = Spec A and f : Y −→X for the structure morphism. We want to apply 5.9 on p : Y −→
Spec A . Notice that Y isquasi-compact and has affine diagonal. Moreover by 3.34 Γ E ≃ Ω A E = Hom ( E , A ) = Hom ( E , f ∗ O Y ) ≃ Hom ( f ∗ E , O Y ) = H (( f ∗ E ) ∨ ) In particular, since Γ O X ≃ A , the map A −→ H ( A ) is an isomorphism. We show now that, ifSpec k −→ Spec A is a geometric point, then Y × A k = ∅ . If g : X k −→ X A is the projection then Y × A k ≃ Spec ( g ∗ A ) , while by 2.12 we have g ∗ A ≃ A Γ ⊗ A k, C . By 3.34 we get Γ ⊗ A k ≃ Ω A Γ ⊗ Ak, C and therefore A Γ ⊗ A k, C = 0 , that is Y × A k = ∅ , implies Γ ⊗ A k = 0 , while Γ O X ⊗ A k = A ⊗ A k ≃ k .It remains to show that {O Y } generates QCoh Y . Since Y f −−→ X is affine, f ∗ C generatesQCoh Y by 5.11. Thus we have to prove that all the sheaves f ∗ E are generated by globalsections. By hypothesis the mapH ( f ∗ E ′∨ ) ⊗ H ( f ∗ E ∨ ) −→ H ( f ∗ ( E ′ ⊗ E ) ∨ ) is an isomorphism for all E , E ′ ∈ C . Since C has duals, choosing E ′ = E ∨ the above map becamethe evaluation ω : H ( f ∗ E ) ⊗ Hom Y ( f ∗ E , O Y ) −→ End Y ( f ∗ E ) Since ω is an isomorphism there exist x , . . . , x n ∈ H ( f ∗ E ) , φ , . . . , φ n ∈ Hom ( f ∗ E , O X ) suchthat id f ∗ E = ω ( P i x i ⊗ φ i ) . This implies that the map O n Y −→ f ∗ E given by the global sections x , . . . , x n is surjective, as required.For the last statement we claim that − ∨ : Fib X , C ( A ) −→ SMex R ( C , A ) , Γ Γ ∨ is an equiv-alence. Taking into account 3.26, if Γ ∈ Fib X , C ( A ) then Γ ∨ ∈ SMex R ( C , A ) because Γ is exacton all finite test sequences and the dual of a right exact sequence of locally free sheaves is againexact. Moreover the map X ( A ) −→ SMex R ( C , A ) factors as X ( A ) P C −−→ Fib X , C ( A ) − ∨ −−→ SMex R ( C , A ) HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 33
Thus − ∨ : Fib X , C ( A ) −→ SMex R ( C , A ) is essentially surjective and − ∨ : SMex R ( C , A ) −→ Fib X , C ( A ) is a quasi-inverse. (cid:3) Proof. (of Theorem 5.6) Let X = Spec B be an affine scheme, { U i } i ∈ I be a set of quasi-compactopen subsets of X and set U = ∩ i U i . If i ∈ I the subscheme U i is the complement of thezero locus of finitely many element of B and thus there exists a free B -module E i and a map φ i : E i −→ B such that U i is the locus where φ i is surjective. Let V : Aff /B −→ ( Sets ) be thefunctor V ( A ) = { ( s i ) i ∈ I | s i ∈ E i ⊗ B A | φ i ( s i ) = 1 } which is an affine scheme. The map V −→ Spec B factors through U and V −→ U is surjective(as functors). Moreover if Spec A −→ U is a map then V × U A = V × X A because U −→ X is amonomorphism. Since V × X A is isomorphic to ( Q i Ker φ i ) × X A we can conclude that V −→ U is an affine fpqc epimorphism, so that U is quasi-compact, and that U −→ X is flat. The resultthen follows from 5.9.For the converse, denote by Z the intersection in the statement, set C = {O U } , B = H ( O U ) and let α : Spec A −→ Spec B be a map which factors through Z and T ∗ : O nU −→ O mU −→O U −→ be an exact sequence on U . Since C = {O Spec B } as monoidal categories, by 5.3 itis enough to show that α ∗|C : C −→
Loc A is exact on T ∗ . The sequence T ∗ defines a complex W ∗ of free A -modules, namely W ∗ = H ( T ∗ ) , and the locus W in Spec A where W ∗ is exact isquasi-compact, open and contains U . Thus Z ⊆ W , the sequence W ∗ become exact on Z andtherefore α ∗ maintains its exactness, as required. (cid:3) Lemma 5.14.
Let R be a set, f : R −→ N be a map and set GL f = Y i ∈R GL f ( i ) and F i for the locally free sheaf of rank f ( i ) on B GL f coming from the universal one on B GL f ( i ) .Then the subcategory of Loc ( B GL f ) consisting of all tensor products of sheaves {F i } i ∈R , { ( Sym m F i ) ∨ } i ∈R s.t. f ( i ) / ∈ R ∗ , m ∈ N generates QCoh ( B GL f ) .Proof. We are going to apply 4.2. Let D be the subcategory of Loc B GL f generated by directsums and tensor products of the sheaves Sym m F i , (det F i ) − for i ∈ R , m ∈ N . We claim that R [ GL f ] is a direct limit of representations in D . The representation R [ GL f ] is a direct limit oftensor products of the regular representations R [ GL f ( i ) ] for i ∈ R . This allows us to reduce to thecase of GL n (i.e. R has one element). Call F the universal rank n sheaf on B GL n and G the free R -module of rank n . As usual we can write R [ GL n ] = R [ X uv ] det for ≤ u, v ≤ n , where det is thedeterminant polynomial. The R -submodule generated by the X u,v is a GL n -subrepresentationisomorphic to F ⊗ G . In particular we obtain injective mapsSym nm ( F ⊗ G ) ⊗ (det F ) −⊗ m −→ R [ GL n ] whose image is the set of fractions f / det m where f is homogeneous of degree nm . Those imagesform an incresing sequence of sub representations saturating R [ GL n ] . Thus R [ GL n ] is the directlimit of the sheaves Sym nm ( F ⊗ G ) ⊗ (det F ) −⊗ m which belongs to D .Coming back to the general framework, by 4.2 and the existence of surjective maps F ⊗ f ( i ) i −→ det F i and, if f ( i ) ∈ R ∗ so that GL f ( i ) is linearly reductive, ( F ∨ i ) ⊗ m −→ ( Sym m F i ) ∨ we get thedesired result. (cid:3) Proof. (of Theorem 5.7). It is easy to see that Fib X , C is a stack (not necessarily in groupoids)for the fpqc topology on Aff /R . To avoid problems with disjoint unions we can assume that X HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 34 is a Zariski stack. We will use notation from 5.14. Let R be a set of representatives of C / ≃ with O X ∈ R , J be a finite subset of R and denote by I J the set of f ∈ N J extending to a function of I . Given f ∈ N J we denote by X f the open locus of X where rk E = f ( E ) for all E ∈ J . Noticethat X = F f ∈I J X f and, since X is quasi-compact, I J is finite. The sheaves ( E |X f ) E∈ J inducesa map X f −→ B GL f = B f and thus a map ω J : X −→ G f ∈I J B f = B J For all
E ∈ J there is a locally free sheaf H E ,J on B J such that ( H E ,J ) |B f is the canonical locallyfree sheaf of rank f ( E ) (pullback from B GL f ( E ) ), so that ω ∗ J H E ,J ≃ E . Let D J be the subcategoryof Loc ( B J ) consisting of all tensor products and duals of H E ,J and, if there exists f ∈ I J such that f ( E ) / ∈ R ∗ , also of Sym n H E ,J for n ∈ N and E ∈ J . We have ω ∗ J D J ⊆ C by construction, that D J generates QCoh ( B J ) by 5.14 and, in particular, that P D J : B J −→ Fib B J , D J is an equivalenceof stacks by 5.3. When J = {E} we will replace J by E in the subscripts.We now show that Fib X , C is a stack in groupoids. If Γ , Γ ′ ∈ Fib X , C ( A ) , δ : Γ −→ Γ ′ is amorphism and E ∈ C then δ E : Γ E −→ Γ ′E is an isomorphism because Γ ◦ ω ∗E δ ◦ ω ∗E −−−→ Γ ′ ◦ ω ∗E is amorphism in the groupoid Fib B E , D E ( A ) ≃ B E ( A ) .Given f : C −→ N , we show that Fib f X , C = ∅ if and only if f ∈ I . For the if part, if f ∈ I thereexists s : Spec L −→ X such that rk s ∗ E = f ( E ) for all E ∈ C . Thus s ∗|C ∈ Fib f X , C . For the onlyif part we show that, if L is an algebraically closed field and Γ ∈ Fib X , C ( L ) then rk Γ ∗ : C −→ N belongs to I . This will also show that, if I is finite, then Fib X , C is the disjoint union of theFib f X , C for f ∈ I : indeed Fib f X , C is the locus where rk G E = f ( E ) for all E ∈ C .So let Γ ∈ Fib X , C ( L ) . Given J ⊆ R finite consider Γ ◦ ω ∗ J ∈ Fib B J , D J ( L ) ≃ B J ( L ) , so thatthere exists a map Spec L s −−→ B J such that Γ ◦ ω ∗ J ≃ s ∗|D J . The map s has image in somecomponent B f with f ∈ I J . In particular if E ∈ J we haverk Γ E = rk Γ ω ∗ J H E ,J = rk ( s ∗ ( H E ,J ) |B f ) = f ( E ) This shows that for all finite subsets J ⊆ R we have that f J = ( rk Γ ∗ ) | J belongs to I J . Byconstruction X f J is a non-empty (open and) closed substack of X . Since X f J ⊆ X f J ′ if J ′ ⊆ J and X is quasi-compact it follows that T J X f J = ∅ and thus that rk Γ ∗ ∈ I .It remains to show the claims about Fib f X , C when f ∈ I . The sheaf G E has rank f ( E ) onFib f X , C and thus we get a map g : Fib f X , C −→ B GL f such that g ∗ F E ≃ G E for E ∈ R (herewe are using notation from 5.14). Let C ′ be the subcategory of Loc B GL f obtained by takingtensor products and duals of the sheaves F E and, if f ( E ) / ∈ R ∗ , also of Sym n F E for n ∈ N and E ∈ R . Moreover set D = {G E } E∈C ⊆ Loc ( Fib X , C ) . We claim that g ∗ C ′ ⊆ D | Fib f X , C . It sufficesto show that if E ∈ C then G E ∨ ≃ ( G E ) ∨ and, if f ( E ) / ∈ R ∗ , G Sym n E ≃ Sym n G E over the wholeFib X , C . Consider J = {E} ⊆ R . The functor ω E : X −→ B E induces ω ∗E : D E −→ C and a functor δ : Fib X , C −→ Fib B E , D E ≃ B E . If t is either ( − ) ∨ or Sym n and Γ ∈ Fib X , C we have isomorphisms G t ( E ) (Γ) = Γ t ( E ) ≃ (Γ ◦ ω ∗E ) t ( H E , {E} ) ≃ δ (Γ) ∗ t ( H E , {E} ) ≃ t ( δ (Γ) ∗ H E , {E} ) ≃ t (Γ E ) = t ( G E )(Γ) natural in Γ and thus that G t ( E ) ≃ t ( G E ) .Using 5.14 and 5.11 all the claims in the statement follow if we show that the fiber product U of g : Fib f X , C −→ B GL f along the canonical map Spec R −→ B GL f is pseudo-affine. SinceLoc ( X ) is essentially small, it is easy to find a sub monoidal subcategory ˜ C of C which is small andsuch that e C −→ C is an equivalence. Via restriction we get an equivalence Fib X , C −→ Fib X , ˜ C andwe can assume that C is small. If A is an R -algebra then U ( A ) is the groupoid of Γ ∈ Fib f X , C ( A ) together with basis of Γ E for all E ∈ C . In particular it is easy to see that U is (equivalent to) a HEAFIFICATION FUNCTORS AND TANNAKA’S RECONSTRUCTION 35 sheaf. Let V : Aff /R −→ ( Sets ) be the functor which maps an R -algebra A to the set of R -linearand strong monoidal functors Γ :
C −→
Loc A such that rk Γ E = f ( E ) together with a basis of Γ E for E ∈ C . The sheaf V is affine because it is a closed subscheme of V = Y E−→E ′ ∈ Arr ( C ) Hom ( R f ( E ) , R f ( E ′ ) ) × Y E , E ′ ∈C Iso ( R f ( E ) ⊗ R f ( E ′ ) , R f ( E⊗E ′ ) ) × G m,R : Aff /R −→ ( Sets ) which is affine. Write V = Spec B and denote by Γ :
C −→
Loc B the canonical R -linear andstrong monoidal functor. Given a finite test sequence T in C the sequence of maps Γ T is acomplex of free B -modules and denote by V T the locus in V where this complex is exact. Clearly U is the intersection of the V T for all finite test sequences T . Since one can easily check that V T is a quasi-compact open subscheme of V the result follows from 5.6. (cid:3) References [BC12] Martin Brandenburg and Alexandru Chirvasitu,
Tensor functors between categories of quasi-coherentsheaves , arxiv:1202.5147 (2012), 21.[Bha14] Bhargav Bhatt,
Algebraization and Tannaka duality , arxiv:1404.7483 (2014), 35.[Bra14] Martin Brandenburg,
Tensor categorical foundations of algebraic geometry , Ph.D. thesis, 2014, p. 251.[Bro13] Michael Broshi, G -torsors over a Dedekind scheme , Journal of Pure and Applied Algebra (2013),no. 1, 11–19.[CLS11] David A. Cox, John B. Little, and Henry K. Schenck, Toric Varieties , American Mathematical Society,2011.[DM82] Pierre Deligne and James S. Milne,
Tannakian Categories , Lecture Notes in Mathematics, vol. 900,Springer-Verlag, Berlin, 1982.[Gro13] Philipp Gross,
Tensor generators on schemes and stacks , arXiv:1306.5418 (2013), 21.[Jan87] Jens Carsten Jantzen,
Representations of algebraic groups , Pure and Applied Mathematics, vol. 131,Academic Press Inc., 1987.[Kaj98] Takeshi Kajiwara,
The functor of a toric variety with enough invariant effective Cartier divisors , To-hoku Mathematical Journal (1998), no. 1, 139–157 (EN).[Lur04] Jacob Lurie, Tannaka Duality for Geometric Stacks , arXiv:math/0412266 (2004), 14.[MBL99] Laurent Moret-Bailly and Gerard Laumon,
Champs algébriques , first ed., Springer, 1999.[Sav06] Valentin Savin,
Tannaka duality on quotient stacks , Manuscripta Mathematica (2006), no. 3, 287–303.[Sch12] Daniel Schäppi,
A characterization of categories of coherent sheaves of certain algebraic stacks ,arXiv:1206.2764 (2012).[Sch13] ,
The formal theory of Tannaka duality , Astérisque (2013), viii+140.[SP014]
Stacks Project , 2014.[Ton13a] Fabio Tonini,
Stacks of Ramified Covers Under Diagonalizable Group Schemes , International Mathe-matics Research Notices (2013), 80.[Ton13b] ,
Stacks of ramified Galois covers , Ph.D. thesis (2013), 192.[Tot04] Burt Totaro,
The resolution property for schemes and stacks , Journal für die Reine und AngewandteMathematik (2004), 23.(Tonini)
Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany
E-mail address ::