Coherent Tannaka duality and algebraicity of Hom-stacks
aa r X i v : . [ m a t h . AG ] J un COHERENT TANNAKA DUALITY AND ALGEBRAICITYOF HOM-STACKS
JACK HALL AND DAVID RYDH
Abstract.
We establish Tannaka duality for noetherian algebraic stackswith affine stabilizer groups. Our main application is the existence ofHom-stacks in great generality. Introduction
Classically, Tannaka duality reconstructs a group from its category offinite-dimensional representations [Tan39]. Various incarnations of Tannakaduality have been studied for decades. The focus of this article is a recentformulation for algebraic stacks [Lur04] which we now recall.Let X be a noetherian algebraic stack. We denote its abelian categoryof coherent sheaves by Coh ( X ). If f : T → X is a morphism of noetherianalgebraic stacks, then there is an induced pullback functor f ∗ : Coh ( X ) → Coh ( T ) . It is well-known that f ∗ has the following three properties:(i) f ∗ sends O X to O T ,(ii) f ∗ preserves the tensor product of coherent sheaves, and(iii) f ∗ is a right exact functor of abelian categories.Hence, there is a functorHom( T, X ) → Hom r ⊗ , ≃ (cid:0) Coh ( X ) , Coh ( T ) (cid:1) , ( f : T → X ) (cid:0) f ∗ : Coh ( X ) → Coh ( T ) (cid:1) , where the right hand side denotes the category with objects the functors F : Coh ( X ) → Coh ( T ) satisfying conditions (i)–(iii) above and morphismsgiven by natural isomorphisms of functors.If X has affine diagonal (e.g., X is the quotient of a variety by an affinealgebraic group), then the functor above is known [Lur04] to be fully faithfulwith image consisting of tame functors. Even though tameness of a functoris a difficult condition to verify, Lurie was able to establish some strikingapplications to algebraization problems.Various stacks of singular curves [AK14, § Date : June 29, 2016.2010
Mathematics Subject Classification.
Primary 14A20; Secondary 14D23, 18D10.
Key words and phrases.
Tannaka duality, algebraic stacks, Hom-stacks, Mayer–Vietorissquares, formal gluings.The first author is supported by the Australian Research Council DE150101799.This collaboration was supported by the G¨oran Gustafsson foundation. The secondauthor is also supported by the Swedish Research Council 2011-5599 and 2015-05554. applications in moduli theory, the results of [Lur04] are insufficient. Themain result of this article is the following theorem, which besides removingLurie’s hypothesis of affine diagonal, obviates tameness.
Theorem 1.1.
Let X be a noetherian algebraic stack with affine stabilizers.For every locally excellent algebraic stack T , the functor Hom(
T, X ) → Hom r ⊗ , ≃ (cid:0) Coh ( X ) , Coh ( T ) (cid:1) is an equivalence. That X has affine stabilizers means that Aut( x ) is affine for every field k and point x : Spec k → X ; equivalently, the diagonal of X has affine fibers.An algebraic stack is locally excellent if there exists a smooth presentationby an excellent scheme (see Remark 7.2); this includes every algebraic stackthat is locally of finite type over a field, over Z , or over a complete localnoetherian ring. We also wish to emphasize that we do not assume that thediagonal of X is separated in Theorem 1.1. The restriction to stacks withaffine stabilizers is a necessary condition for the equivalence in Theorem 1.1(see Theorem 10.1).Theorem 1.1 is a consequence of Theorem 8.4, which also gives variousrefinements in the non-noetherian situation and when X has quasi-affine orquasi-finite diagonal. Main applications.
In work with J. Alper [AHR15], Theorem 1.1 is ap-plied to resolve Alper’s conjecture on the local quotient structure of algebraicstacks [Alp10]. A more immediate application of Theorem 1.1 is the follow-ing algebraicity result for Hom-stacks, generalizing all previously knownresults and answering [AOV11, Question 1.4].
Theorem 1.2.
Let Z → S and X → S be morphisms of algebraic stackssuch that Z → S is proper and flat of finite presentation, and X → S is locally of finite presentation, quasi-separated, and has affine stabilizers.Then (i) the stack Hom S ( Z, X ) : T Hom S ( Z × S T, X ) , is algebraic; (ii) the morphism Hom S ( Z, X ) → S is locally of finite presentation,quasi-separated, and has affine stabilizers; and (iii) if X → S has affine (resp. quasi-affine, resp. separated) diagonal,then so has Hom S ( Z, X ) → S . Theorem 1.2 has already seen applications to log geometry [Wis16], anarea which provides a continual source of stacks that are neither Deligne–Mumford nor have separated diagonals. In general, the condition that X has affine stabilizers is necessary (see Theorem 10.4).There are analogous algebraicity results for Weil restrictions (that is, re-strictions of scalars). Theorem 1.3.
Let f : Z → S and g : X → Z be morphisms of algebraicstacks such that f is proper and flat of finite presentation and f ◦ g is locallyof finite presentation, quasi-separated and has affine stabilizers. Then (i) the stack f ∗ X = R Z/S ( X ) : T Hom Z ( Z × S T, X ) is algebraic; (ii) the morphism R Z/S ( X ) → S is locally of finite presentation, quasi-separated and has affine stabilizers; and ANNAKA DUALITY 3 (iii) if g has affine (resp. quasi-affine, resp. separated) diagonal, then sohas f ∗ X → S . When Z has finite diagonal and X has quasi-finite and separated diagonal,Theorems 1.2 and 1.3 were proved in [HR14, Thms. 3 & 4]. In Corollary 9.2,we also excise the finite presentation assumptions on X → S in Theorems1.2 and 1.3, generalizing the results of [HR15b, Thm. 2.3 & Cor. 2.4] forstacks with quasi-finite diagonal. Application to descent. If X has quasi-affine diagonal, then it is well-known that it is a stack for the fpqc topology [LMB, Cor. 10.7]. In general, itis only known that algebraic stacks satisfy effective descent for fppf coverings.Nonetheless, using that QCoh is a stack for the fpqc-topology and Tannakaduality, we are able to establish the following result.
Corollary 1.4.
Let X be a quasi-separated algebraic stack with affine sta-bilizers. Let π : T ′ → T be an fpqc covering such that T is a locally excellentstack and T ′ is locally noetherian. Then X satisfies effective descent for π . Application to completions.
Another application concerns completions.
Corollary 1.5.
Let A be a noetherian ring and let I ⊆ A be an ideal.Assume that A is complete with respect to the I -adic topology. Let X be anoetherian algebraic stack and consider the natural morphism X ( A ) → lim ←− X ( A/I n ) of groupoids. This morphism is an equivalence if either (i) X has affine stabilizers and A is excellent (or merely a G-ring); or (ii) X has quasi-affine diagonal; or (iii) X has quasi-finite diagonal. Using derived methods, Corollary 1.5(ii) was recently proved for non-noetherian complete rings A [BH15, Bha14]. That X has affine stabilizersin Corollary 1.5 is necessary (see Theorem 10.5). On the proof of Tannaka duality.
We will discuss the proof of Theorem8.4, the refinement of Theorem 1.1. The reason for this is that it is muchmore convenient from a technical standpoint to consider the problem in thesetting of quasi-coherent sheaves on potentially non-noetherian algebraicstacks.So let T and X be algebraic stacks and let QCoh ( T ) and QCoh ( X ) denotetheir respective abelian categories of quasi-coherent sheaves. We will assumethat X is quasi-compact and quasi-separated. Our principal concern is theproperties of the functor ω X ( T ) : Hom( T, X ) → Hom c ⊗ ( QCoh ( X ) , QCoh ( T )) , ( f : T → X ) ( f ∗ : QCoh ( X ) → QCoh ( T )) , where the right hand side denotes the additive functors F : QCoh ( X ) → QCoh ( T ) satisfying(i) F ( O X ) = O T ,(ii) F preserves the tensor product, and(iii) F is right exact and preserve (small) direct sums. JACK HALL AND DAVID RYDH
We call such F cocontinuous tensor functors .An algebraic stack X has the resolution property if every quasi-coherentsheaf is a quotient of a direct sum of vector bundles. In Theorem 4.10we establish the equivalence of ω X ( T ) when X has affine diagonal and theresolution property. This result has appeared in various forms in the workof others (cf. Sch¨appi [Sch12, Thm. 1.3.2], Savin [Sav06] and Brandenburg[Bra14, Cor. 5.7.12]) and forms an essential stepping stone in the proof ofour main theorem (Theorem 8.4).In general, there are stacks—even schemes—that do not have the resolu-tion property. Indeed, if X has the resolution property, then X has at leastaffine diagonal [Tot04, Prop. 1.3]. Our proof uses the following three ideasto overcome this problem:(i) If U ⊆ X is a quasi-compact open immersion and QCoh ( X ) → QCoh ( T ) is a tensor functor, then there is an induced tensor func-tor QCoh ( U ) → QCoh ( V ) where V ⊆ T is the “inverse image of U ”. The proof of this is based on ideas of Brandenburg and Chirv-asitu [BC14]. (Section 5)(ii) If X is an infinitesimal neighborhood of a stack with the resolutionproperty, then ω X ( T ) is an equivalence for all T . (Section 6)(iii) There is a constructible stratification of X into stacks with affinediagonal and the resolution property (Proposition 8.2). We deducethe main theorem by induction on the number of strata using formalgluings [MB96, HR16]. This step uses special cases of Corollaries 1.4and 1.5. (Sections 7 and 8)In the third step, we assume that our functors preserve sheaves of finitetype. Open questions.
Concerning (ii), it should be noted that we do not knowthe answers to the following two questions.
Question 1.6. If X has the resolution property and X ֒ → X is a nilpotentclosed immersion, then does X have the resolution property? The question has an affirmative answer if X is cohomologically affine,e.g., X = B k G where G is a linearly reductive group scheme over k . Thequestion is open if X = B k G where G is not linearly reductive, even if X = B k [ ǫ ] G ǫ where G ǫ is a deformation of G over the dual numbers [Con10]. Question 1.7. If X ֒ → X is a nilpotent closed immersion and ω X ( T ) isan equivalence, is then ω X ( T ) an equivalence? Step (ii) answers neither of these questions but uses a special case of thefirst question (Lemma 6.2) and the conclusion (Main Lemma 6.1) is a specialcase of the second question.The following technical question also arose in this research.
Question 1.8.
Let X be an algebraic stack with quasi-compact and quasi-separated diagonal and affine stabilizers. Let k be a field. Is every morphism Spec k → X affine? If X ´etale-locally has quasi-affine diagonal, then Question 1.8 has an affir-mative answer (Lemma 4.6). This makes finding counterexamples extraor-dinarily difficult and thus very interesting. This question arose because ANNAKA DUALITY 5 if Spec k → X is non-affine, then ω X (Spec k ) is not fully faithful (Theo-rem 10.2). This explains our restriction to natural isomorphisms in Theo-rem 1.1. Note that every morphism Spec k → X as in Question 1.8 is atleast quasi-affine [Ryd11a, Thm. B.2]. We do not know the answer to thequestion even if X has separated diagonal and is of finite type over a field. On the applications.
Let T be a noetherian and locally excellent algebraicstack and let Z be a closed substack defined by a coherent ideal J ⊆ O T .Let Z [ n ] be the closed substack defined by J n +1 . Assume that the naturalfunctor Coh ( T ) → lim ←− n Coh ( Z [ n ] ) is an equivalence of categories. Then animmediate consequence of Tannaka duality (Theorem 1.1) is thatHom( T, X ) → lim ←− Hom( Z [ n ] , X )is an equivalence of categories for every noetherian algebraic stack X withaffine stabilizers. This applies in particular if A is excellent and I -adicallycomplete and T = Spec A and Z = Spec A/I ; this gives Corollary 1.5. Moregenerally, it also applies if T is proper over Spec A and Z = T × Spec A Spec
A/I (Grothendieck’s existence theorem). This latter case is fed into Artin’s cri-terion to prove Theorem 1.2 (the remaining hypotheses have largely beenverified elsewhere).There are also non-proper stacks T satisfying Coh ( T ) → lim ←− n Coh ( Z [ n ] ),such as global quotient stacks with proper good moduli spaces (see [GZB15,AB05] for some special cases). This featured in the resolution of Alper’sconjecture [AHR15].Such statements, and their derived versions, were also recently consideredby Halpern-Leistner–Preygel [HP14]. There, they considered variants of ourTheorem 1.2. For their algebraicity results, their assumption was similar toassuming that Coh ( T ) → lim ←− n Coh ( Z [ n ] ) was an equivalence (though theyalso considered other derived versions), and that X → S was locally of finitepresentation with affine diagonal. Relation to other work.
As mentioned in the beginning of the Introduc-tion, Lurie identifies the image of ω X ( T ) with the tame functors when X isquasi-compact with affine diagonal [Lur04]. Tameness means that faithfulflatness of objects is preserved. This is a very strong assumption that makesit possible to directly pull back a smooth presentation of X to a smoothcovering of T and deduce the result by descent. Note that every tensorfunctor preserves coherent flat objects—these are vector bundles and hencedualizable—but this does not imply that flatness of quasi-coherent objectsare preserved. Lurie’s methods also work for non-noetherian T .Brandenburg and Chirvasitu have shown that ω X ( T ) is an equivalence forevery quasi-compact and quasi-separated scheme X [BC14], also for non-noetherian T . The key idea of their proof is the tensor localization that wehave adapted in Section 5. Using this technique, we give a slightly simplifiedproof of their theorem in Theorem 5.10.When X has quasi-affine diagonal, derived variants of Theorem 1.1 haverecently been considered by various authors [FI13, Bha14, BH15]. Specifi-cally, they were concerned with symmetric monoidal ∞ -functors G : D ( X ) → D ( T ) between stable ∞ -categories of quasi-coherent sheaves. JACK HALL AND DAVID RYDH
It is not obvious how to go from a tensor functor F : QCoh ( X ) → QCoh ( T )to a symmetric monoidal ∞ -functor L F : D ( X ) → D ( T ), so our results can-not be deduced from the derived perspective. When T is locally noetherian,however, our result is stronger than [BH15, Thm. 1.4]. Indeed, the functors G : D ( X ) → D ( T ) are assumed to preserve derived tensor products, connec-tive complexes (i.e., are right t -exact) and pseudo-coherent complexes andhence induces a right-exact tensor functor H ( G ) : QCoh ( X ) → QCoh ( T )preserving sheaves of finite type. When X has finite stabilizers, the right t -exactness is sometimes automatic [FI13, Bha14, BZ10].We do not address the Tannaka recognition problem, i.e., which symmet-ric monoidal categories arise as the category of quasi-coherent sheaves onan algebraic stack. For gerbes, this has been done in characteristic zero byDeligne [Del90, Thm. 7.1]. For stacks with the resolution property, this hasbeen done by Sch¨appi [Sch14, Thm. 1.4], [Sch15, Thms. 1.2.2, 5.3.10]. Simi-lar results from the derived perspective have been considered by Wallbridge[Wal12] and Iwanari [Iwa14]. Acknowledgments.
We would like to thank Dan Abramovich for asking uswhether Theorem 1.2 held, which was the original motivation for this article.We would also like to thank Martin Brandenburg for his many comments ona preliminary draft and for sharing his dissertation with us. In particular,he made us aware of Proposition 3.5, which greatly simplified Corollary 3.6.Finally, we would like to thank Bhargav Bhatt for several useful commentsand suggestions, Jarod Alper for several interesting and supportive discus-sions, and Andrew Kresch for some encouraging remarks.2.
Symmetric monoidal categories A symmetric monoidal category is the data of a category C , a tensorproduct ⊗ C : C × C → C , and a unit O C that together satisfy variousnaturality, commutativity, and associativity properties [ML98, VII.7]. Asymmetric monoidal category C is closed if for any M ∈ C the functor − ⊗ C M : C → C admits a right adjoint, which we denote as H om C ( M, − ). Example 2.1.
Let A be a ring; then the category of A -modules, Mod ( A ),together with its tensor product ⊗ A , is a symmetric monoidal category withunit A . In fact, Mod ( A ) is even closed: the right adjoint to − ⊗ A M is the A -module Hom A ( M, − ). If A is noetherian, then the subcategory of finite A -modules, Coh ( A ), is also a closed symmetric monoidal category.A functor F : C → D between symmetric monoidal categories is laxsymmetric monoidal if for each M and M ′ of C there are natural maps F ( M ) ⊗ D F ( M ′ ) → F ( M ⊗ C M ′ ) and O D → F ( O C ) that are compatiblewith the symmetric monoidal structure. If these maps are both isomor-phisms, then F is symmetric monoidal . Note that if F : C → D is a sym-metric monoidal functor, then a right adjoint G : D → C to F is always laxsymmetric monoidal. Example 2.2.
Let φ : A → B be a ring homomorphism. The functor − ⊗ A B : Mod ( A ) → Mod ( B ) is symmetric monoidal. It admits a right adjoint, Mod ( B ) → Mod ( A ), which is given by the forgetful functor. This forgetfulfunctor is lax monoidal, but not monoidal. ANNAKA DUALITY 7 If C is a symmetric monoidal category, then a commutative C -algebra consists of an object A of C together with a multiplication m : A ⊗ C A → A and a unit e A : O C → A with the expected properties [ML98, VII.3]. Let CAlg ( C ) denote the category of commutative C -algebras. The category CAlg ( C ) is naturally endowed with a symmetric monoidal structure thatmakes the forgetful functor CAlg ( C ) → C symmetric monoidal. Example 2.3. If A is a ring, then CAlg ( Mod ( A )) is the category of com-mutative A -algebras.The following observation will be used frequently: if F : C → D is a laxsymmetric monoidal functor and A is a commutative C -algebra, then F ( A )is a commutative D -algebra.3. Abelian tensor categories An abelian tensor category is a symmetric monoidal category that isabelian and the tensor product is right exact and preserves finite directsums in each variable (i.e., preserves all finite colimits in each variable).Recall that an abelian category is Grothendieck if it is closed under smalldirect sums, filtered colimits are exact, and it has a generator [Stacks, Tag079A]. Also, recall that a functor F : C → D between two Grothendieckabelian categories is cocontinuous if it is right-exact and preserves smalldirect sums, equivalently, it preserves all small colimits.A Grothendieck abelian tensor category is an abelian tensor category suchthat the underlying abelian category is Grothendieck abelian and the tensorproduct is cocontinuous in each variable. By the Special Adjoint FunctorTheorem [KS06, Prop. 8.3.27(iii)], if C is a Grothendieck abelian tensorcategory, then it is also closed. Example 3.1.
Let A be a ring. Then Mod ( A ) is a Grothendieck abeliantensor category. If A is noetherian, then Coh ( A ) is an abelian tensor categorybut not Grothendieck abelian—it is not closed under small direct sums.A tensor functor F : C → D is an additive symmetric monoidal func-tor between abelian tensor categories. Let GTC be the 2-category ofGrothendieck abelian tensor categories and cocontinuous tensor functors.By the Special Adjoint Functor Theorem, if F : C → D is a cocontinuoustensor functor, then F admits a (lax symmetric monoidal) right adjoint. Example 3.2.
Let T be a ringed site. The category of O T -modules Mod ( T )is a Grothendieck abelian tensor category with unit O T and the internal Homis the functor H om O T ( M, − ). Example 3.3.
Let X be an algebraic stack. The category of quasi-coherentsheaves QCoh ( X ) is a Grothendieck abelian tensor category with unit O X [Stacks, Tag 0781]. The internal Hom is QC( H om O X ( M, − )), where QCdenotes the quasi-coherator (the right adjoint to the inclusion of the categoryof quasi-coherent sheaves in the category of lisse-´etale O X -modules). If X isan algebraic stack, then CAlg ( QCoh ( X )) is the symmetric monoidal categoryof quasi-coherent O X -algebras.If f : X → Y is a morphism of algebraic stacks, then the resulting functor f ∗ : QCoh ( Y ) → QCoh ( X ) is a cocontinuous tensor functor. If f is flat, then JACK HALL AND DAVID RYDH f ∗ is exact. We always denote the right adjoint of f ∗ by f ∗ : QCoh ( X ) → QCoh ( Y ). If f is quasi-compact and quasi-separated, then f ∗ coincides withthe pushforward of lisse-´etale O X -modules [Ols07, Lem. 6.5(i)]. In par-ticular, if f is quasi-compact and quasi-separated, then f ∗ : QCoh ( X ) → QCoh ( Y ) preserves directed colimits (work smooth-locally on Y and thenapply [Stacks, Tag 0738]) and is lax symmetric monoidal. Definition 3.4.
Given abelian tensor categories C and D , we let Hom c ⊗ ( C , D )(resp. Hom r ⊗ ( C , D )) denote the category of cocontinuous (resp. right ex-act) tensor functors and natural transformations. The transformations arerequired to be natural with respect to both homomorphisms and the sym-metric monoidal structure. We let Hom c ⊗ , ≃ ( C , D ) (resp. Hom r ⊗ , ≃ ( C , D ))denote the groupoid of cocontinuous (resp. right exact) tensor functors andnatural isomorphisms.We conclude this section with some useful facts for the paper. We firstconsider modules over algebras, which are addressed, for example, in Bran-denburg’s thesis [Bra14, § Modules over an algebra in tensor categories.
Let C be a Grothen-dieck abelian tensor category and let A be a commutative C -algebra. De-fine Mod C ( A ) to be the category of A -modules. Objects are pairs ( M, a ),where M ∈ C and a : A ⊗ C M → M is an action of A on M . Morphisms φ : ( M, a ) → ( M ′ , a ′ ) in Mod C ( A ) are those C -morphisms φ : M → M ′ thatpreserve the respective actions. We identify A with ( A, m ) ∈ Mod C ( A )where m : A ⊗ C A → A is the multiplication. It is straightforward to showthat Mod C ( A ) is a Grothendieck abelian tensor category, with tensor prod-uct ⊗ A and unit A , and the natural forgetful functor Mod C ( A ) → C pre-serves all limits and colimits [KS06, § s : A → B is a C -algebra homomorphism, then there is a natural co-continuous tensor functor s ∗ : Mod C ( A ) → Mod C ( B ) , ( M, a ) ( B ⊗ A M, B ⊗ A a ) . Suppose f ∗ : C → D is a cocontinuous tensor functor with right adjoint f ∗ : D → C . If A is a commutative C -algebra, then there is a naturalinduced cocontinuous tensor functor f ∗ A : Mod C ( A ) → Mod D ( f ∗ A ) , ( M, a ) ( f ∗ M, f ∗ a ) . Noting that ǫ : f ∗ f ∗ O D → O D is a D -algebra homomorphism, there is anatural induced cocontinuous tensor functor¯ f ∗ : Mod C ( f ∗ O D ) f ∗ f ∗O D −−−−→ Mod D ( f ∗ f ∗ O D ) ǫ ∗ −→ Mod D ( O D ) = D . Moreover, if we let η : O C → f ∗ f ∗ O C = f ∗ O D denote the unit, then f ∗ =¯ f ∗ η ∗ . We have the following striking characterization of module categories[Bra14, Prop. 5.3.1]. Proposition 3.5.
Let C be a Grothendieck abelian tensor category and let A be a commutative algebra in C . Then for every Grothendieck abelian tensor ANNAKA DUALITY 9 category D , there is an equivalence of categories Hom c ⊗ ( Mod C ( A ) , D ) ≃ { ( F, h ) : F ∈ Hom c ⊗ ( C , D ) ,h ∈ Hom
CAlg ( D ) ( F ( A ) , O D ) } , where a morphism ( F, h ) → ( F ′ , h ′ ) is a natural transformation α : F → F ′ such that h = h ′ ◦ α ( A ) . The following corollary is immediate (see [Bra14, Cor. 5.3.7]).
Corollary 3.6.
Let p : Y ′ → Y be an affine morphism of algebraic stacks.Let X be an algebraic stack and let g ∗ : QCoh ( Y ) → QCoh ( X ) be a cocon-tinuous tensor functor. If X ′ is the affine X -scheme S pec X ( g ∗ p ∗ O Y ′ ) withstructure morphism p ′ : X ′ → X , then there is a -cocartesian diagram in GTC : QCoh ( X ′ ) QCoh ( Y ′ ) g ′∗ o o QCoh ( X ) p ′∗ O O QCoh ( Y ) . p ∗ O O g ∗ o o Moreover, the natural transformation g ∗ p ∗ ⇒ p ′∗ g ′∗ is an isomorphism. Note that if g ∗ comes from a morphism g : X → Y , then X ′ ∼ = X × Y Y ′ .3.2. Inverse limits of abelian tensor categories.
We will now brieflydiscuss some inverse limits that will be crucial when we apply Tannakaduality to establish the algebraicity of Hom-stacks in Theorem 1.2. Thefollowing notation will be useful.
Notation 3.7.
Let i : Z → X be a closed immersion of algebraic stacks de-fined by a quasi-coherent ideal I . For each integer n ≥
0, we let i [ n ] : Z [ n ] → X denote the closed immersion defined by the quasi-coherent ideal I n +1 ,which we call the n th infinitesimal neighborhood of Z . Let X be a noetherian algebraic stack and let i : Z → X be a closed im-mersion. Let Coh ( X, Z ) denote the category lim ←− n Coh ( Z [ n ] ). The argumentsof [Stacks, Tag 087X] easily extend to establish the following:(i) Coh ( X, Z ) is an abelian tensor category with unit {O Z [ n ] } and ten-sor product { M n } n ≥ ⊗ { N n } n ≥ = { M n ⊗ O Z [ n ] N n } n ≥ ;(ii) if p : U → X is smooth and quasi-compact, then the restriction Coh ( X, Z ) → Coh ( U, U × X Z ) is an exact tensor functor; and(iii) exactness in Coh ( X, Z ) may be checked on a smooth covering of X .If { f n : M n → N n } n ≥ is a morphism in Coh ( X, Z ), then it is easily de-termined that coker( { f n } n ≥ ) ∼ = { coker f n } n ≥ . Computing ker( { f n } n ≥ ) ismore involved. We will need the following lemma. Lemma 3.8.
Coh ( X, Z ) is the limit of the inverse system of categories { Coh ( Z [ n ] ) } n ≥ in the -category of abelian tensor categories with right exacttensor functors and natural isomorphisms of tensor functors.Proof. It remains to verify that if F n : C → Coh ( Z [ n ] ) is a compatible se-quence of right exact abelian tensor functors, then the induced abelian tensorfunctor lim ←− n F n : C → Coh ( X, Z ) is right exact. The explicit description ofcokernels in
Coh ( X, Z ) shows that this is the case. (cid:3) Tensorial algebraic stacks
Let T and X be algebraic stacks. There is an induced functor ω X ( T ) : Hom( T, X ) → Hom c ⊗ (cid:0) QCoh ( X ) , QCoh ( T ) (cid:1) that takes a morphism f to f ∗ . We also let Hom ft c ⊗ (cid:0) QCoh ( X ) , QCoh ( T ) (cid:1) denote the full subcategory of functors that preserve sheaves of finite type.Similarly, we let Hom c ⊗ , ≃ (cid:0) QCoh ( X ) , QCoh ( T ) (cid:1) denote the subcategory ofnatural isomorphisms of functors. Clearly, ω X ( T ) factors through all ofthese subcategories and we let ω X, ≃ ( T ), ω ft X ( T ) and ω ft X, ≃ ( T ) denote therespective factorizations. Note that when X and T are locally noetherian,the natural functor:Hom r ⊗ (cid:0) Coh ( X ) , Coh ( T ) (cid:1) → Hom ft c ⊗ (cid:0) QCoh ( X ) , QCoh ( T ) (cid:1) is an equivalence of categories. Thus, Theorem 1.1 says that ω ft X, ≃ ( T ) is anequivalence.Since QCoh ( − ) is a stack in the fpqc topology, the target categories ofthe functors ω X , ω X, ≃ , ω ft X and ω ft X, ≃ are stacks in the fpqc topology whenvarying T —for an elaborate proof of this, see [LT12, Thm. 1.1]. The sourcecategories Hom( T, X ) are groupoids and, when varying T , form a stack forthe fppf topology in general and for the fpqc topology when X has quasi-affine diagonal [LMB, Cor. 10.7]. Definition 4.1.
Let T and X be algebraic stacks. We say that a tensorfunctor f ∗ : QCoh ( X ) → QCoh ( T ) is algebraic if it arises from a morphism ofalgebraic stacks f : T → X . If f, g : T → X are morphisms, then a naturaltransformation τ : f ∗ ⇒ g ∗ of tensor functors is realizable if it is induced bya 2-morphism f ⇒ g . We say that X is tensorial if ω X ( T ) is an equivalencefor every algebraic stack T , or equivalently, for every affine scheme T [Bra14,Def. 3.4.4].We begin with a descent lemma. Lemma 4.2.
Let X be an algebraic stack. Let p : T ′ → T be a morphismof algebraic stacks that is covering for the fpqc topology. Let T ′′ = T ′ × T T ′ and T ′′′ = T ′ × T T ′ × T T ′ . Assume that p is a morphism of effective descentfor X (e.g., p is flat and locally of finite presentation). (i) Let f , f : T → X be morphisms and let τ, τ ′ : f ⇒ f be -morphisms. If p ∗ τ = p ∗ τ ′ : f ◦ p ⇒ f ◦ p then τ = τ ′ . (ii) Let f , f : T → X be morphisms and let γ : f ∗ ⇒ f ∗ be a naturaltransformation. If p ∗ γ : p ∗ f ∗ ⇒ p ∗ f ∗ is realizable and ω X ( T ′′ ) isfaithful, then γ is realizable. (iii) Let f ∗ : QCoh ( X ) → QCoh ( T ) be a cocontinuous tensor functor. If p ∗ f ∗ is algebraic, ω X, ≃ ( T ′′ ) is fully faithful and ω X ( T ′′′ ) is faithful,then f ∗ is algebraic.Let ω ∈ { ω X , ω X, ≃ , ω ft X , ω ft X, ≃ } and P ∈ { all , locally noetherian , locally excellent } .If ω ( T ) is faithful (resp. fully faithful, resp. an equivalence) for every affinescheme T with property P , then ω ( T ) is faithful (resp. fully faithful, resp.an equivalence) for every algebraic stack T with property P . ANNAKA DUALITY 11
Proof.
It is sufficient to observe that Hom( − , X ) is a stack in groupoids forthe covering p and Hom c ⊗ ( QCoh ( X ) , QCoh ( − )) is an fpqc stack in categories,so the result boils down to a straightforward and general result for a 1-morphism of such stacks. (cid:3) We next recall two basic lemmas on tensorial stacks. The first is thecombination of [Bra14, Cor. 5.3.4 & 5.6.4].
Lemma 4.3.
Let q : X ′ → X be a quasi-affine morphism of algebraic stacks.If T is an algebraic stack and ω X ( T ) is faithful, fully faithful or an equiva-lence; then so is ω X ′ ( T ) . In particular, if X is tensorial, then so is X ′ .Proof. Since q is the composition of a quasi-compact open immersion fol-lowed by an affine morphism, it suffices to treat these two cases separately.When q is affine the result is an easy consequence of Proposition 3.5. If q is a quasi-compact open immersion, then the counit q ∗ q ∗ → id QCoh ( X ′ ) is anisomorphism; the result now follows from [BC14, Prop. 2.3.6]. (cid:3) The second lemma is well-known (e.g., it is a very special case of [BC14,Thm. 3.4.2]).
Lemma 4.4.
Every quasi-affine scheme is tensorial.Proof.
By Lemma 4.3, it is sufficient to prove that X = Spec Z is tensorial,which is well-known. We refer the interested reader to [BC14, Cor. 2.2.4] or[Bra14, Cor. 5.2.3]. (cid:3) Lacking an answer to Question 1.8 in general, we are forced to make thefollowing definition to treat natural transformations that are not isomor-phisms.
Definition 4.5.
An algebraic stack X is affine-pointed if every morphismSpec k → X , where k is a field, is affine.Note that if X is affine-pointed, then it has affine stabilizers. The follow-ing lemma shows that many algebraic stacks with affine stabilizers that areencountered in practice are affine-pointed. Lemma 4.6.
Let X be an algebraic stack. (i) If X has quasi-affine diagonal, then X is affine-pointed. (ii) Let g : V → X be a quasi-finite and faithfully flat morphism of finitepresentation (not necessarily representable). If V is affine-pointed,then X is affine-pointed.Proof. Throughout, we fix a field k and a morphism x : Spec k → X .For (i), since k is a field, every extension in QCoh (Spec k ) is split; thus x ∗ iscohomologically affine [Alp13, Def. 3.1]. Since X has quasi-affine diagonal,this property is preserved after pulling back x along a smooth morphism p : U → X , where U is an affine scheme [Alp13, Prop. 3.10(vii)]. By Serre’sCriterion [EGA, II.5.2.2], the morphism Spec k × X U → U is affine; and thiscase follows.For (ii), the pullback of g along x gives a quasi-finite and faithfully flatmorphism g : V → Spec k . Since V is discrete with finite stabilizers, thereexists a finite surjective morphism W → V where W is a finite disjoint union of spectra of fields. By assumption W → V → V is affine; hence so is V → V (by Chevalley’s Theorem [Ryd15, Thm. 8.1] applied smooth-locallyon V ). By descent, Spec k → X is affine and the result follows. (cid:3) The following lemma highlights the benefits of affine-pointed stacks.
Lemma 4.7.
Let f , f : T → X be morphisms of algebraic stacks andlet γ : f ∗ ⇒ f ∗ be a natural transformation of cocontinuous tensor func-tors. If X is affine-pointed, then the induced maps of topological spaces | f | , | f | : | T | → | X | coincide.Proof. It suffices to prove that if T = Spec k , where k is a field, then γ isrealizable. Since X is affine-pointed, the morphisms f and f are affine.Also, the natural transformation γ induces, by adjunction, a morphism ofquasi-coherent O X -algebras γ ∨ ( O T ) : ( f ) ∗ O T → ( f ) ∗ O T . In particular, γ ∨ ( O T ) induces a morphism v : T → T over X . We are now free to replace X by T , f by id T , and f by v . Since T is affine, the result now followsfrom Lemma 4.4. (cid:3) We can now prove the following proposition (generalizing Lurie’s corre-sponding result for an algebraic stack with affine diagonal).
Proposition 4.8.
Let X be an algebraic stack. (i) If T is an algebraic stack and X has quasi-affine diagonal, then thefunctor ω X ( T ) is fully faithful. (ii) Let T be a quasi-affine scheme and let f , f : T → X be quasi-affinemorphisms. (a) If α , β : f ⇒ f are -morphisms and α ∗ = β ∗ as naturaltransformations f ∗ ⇒ f ∗ , then α = β . (b) Let γ : f ∗ ⇒ f ∗ be a natural transformation of cocontinuoustensor functors. If γ is an isomorphism or X is affine-pointed,then γ is realizable.Proof. For (i), we may assume that T is an affine scheme (Lemma 4.2).Then every morphism T → X is quasi-affine and the result follows by (ii)and Lemma 4.6(i).For (ii), there are quasi-compact open immersions i k : T ֒ → V k over X ,where V k := S pec X (( f k ) ∗ O T ) and k = 1, 2. Let v k : V k → X be the induced1-morphism.We first treat (ii)(a). The hypotheses imply that α ∗ = β ∗ as naturalisomorphisms of functors from ( f ) ∗ to ( f ) ∗ . In particular, α ∗ and β ∗ inducethe same 1-morphism from V to V over X . Since i and i are openimmersions, they are monomorphisms; hence α = β .We now treat (ii)(b). The natural transformation γ : f ∗ ⇒ f ∗ uniquely in-duces a natural transformation of lax symmetric monoidal functors γ ∨ : ( f ) ∗ ⇒ ( f ) ∗ . In particular, there is an induced morphism of quasi-coherent O X -algebras γ ∨ ( O T ) : ( f ) ∗ O T → ( f ) ∗ O T ; hence a morphism of algebraic stacks g : V → V over X . Note that γ ∨ uniquely induces a natural transforma-tion of lax symmetric monoidal functors ( i ) ∗ ⇒ g ∗ ( i ) ∗ , and by adjunc-tion we have a uniquely induced natural transformation of tensor functors γ ′ : ( g ◦ i ) ∗ ⇒ i ∗ . ANNAKA DUALITY 13
Replacing X by V , f by g ◦ i , f by i , and γ by γ ′ , we may assumethat f is a quasi-compact open immersion such that O X → ( f ) ∗ O T is anisomorphism.If γ is an isomorphism, then f is also a quasi-compact open immersion.Let Z and Z denote closed substacks of X whose complements are f ( T )and f ( T ), respectively. Then f ∗ O Z ∼ = f ∗ O Z ∼ = 0, so f ( T ) ⊆ f ( T ).Arguing similarly, we obtain the reverse inclusion and we see that f ( T ) = f ( T ). Since f and f are open immersions, we obtain the result when γ isassumed to be an isomorphism.Otherwise, Lemma 4.7 implies that f factors through f ( T ) ⊆ X . Wemay now replace X by T and γ with ( f ) ∗ ( γ ) [BC14, Prop. 2.3.6]. Then X is quasi-affine and the result follows from Lemma 4.4. (cid:3) From Proposition 4.8(ii)(b), we obtain an analogue of Lemma 4.7 fornatural isomorphisms of functors when X has affine stabilizers (as opposedto affine-pointed). Corollary 4.9.
Let f , f : T → X be morphisms of algebraic stacks andlet γ : f ∗ ≃ f ∗ be a natural isomorphism of cocontinuous tensor functors. If X has affine stabilizers and quasi-compact diagonal, then the induced mapsof topological spaces | f | , | f | : | T | → | X | coincide.Proof. It suffices to prove the result when T = Spec k , where k is a field.Since X has affine stabilizers and quasi-compact diagonal the morphisms f and f are quasi-affine [Ryd11a, Thm. B.2]. The result now follows fromProposition 4.8(ii)(b). (cid:3) The following result, in a slightly different context, was proved by Sch¨appi[Sch12, Thm. 1.3.2]. Using the Totaro–Gross theorem, we can simplifySch¨appi’s arguments in the algebro-geometric setting.
Theorem 4.10.
Let X be a quasi-compact and quasi-separated algebraicstack with affine stabilizers. If X has the resolution property, then it istensorial.Proof. Let T be an algebraic stack. By Totaro–Gross [Gro13], there is aquasi-affine morphism g : X → BGL N, Z . By Lemma 4.3, it is enough toconsider X = BGL N, Z . Note that X is quasi-compact with affine diago-nal, so the functor ω X ( T ) is fully faithful (Proposition 4.8). It remains toprove that every cocontinuous tensor functor f ∗ : QCoh ( X ) → QCoh ( T ) isalgebraic.Let p : Spec Z → BGL N, Z be the universal GL N -bundle and let A = p ∗ Z be the regular representation. There is an exact sequence0 → O BGL N, Z → A → Q → A as the directed colimit of its sub-sheaves A λ of finite type containing the unit and let Q λ = A λ / O BGL N, Z ⊆ Q .Then A λ and Q λ are vector bundles.It is well-known that (1) any tensor functor f ∗ : QCoh ( X ) → QCoh ( T )preserves dualizable objects and exact sequences of dualizable objects (forexample, see [Bra14, Def. 4.7.1 & Lem. 4.7.10]) and (2) the dualizable ob-jects in QCoh ( Y ) are the vector bundles for any algebraic stack Y [Bra14, Prop. 4.7.5]. We thus have exact sequences0 → O T → f ∗ A λ → f ∗ Q λ → f ∗ is cocontinuous, we also obtain an exact sequence0 → O T → f ∗ A → f ∗ Q → f ∗ A is a faithfully flat algebra.Let V = S pec T ( f ∗ A ); then r : V → T is faithfully flat. By Corollary 3.6,we have a cocartesian diagram QCoh ( V ) QCoh (Spec Z ) f ′∗ o o QCoh ( T ) r ∗ O O QCoh ( X ) . p ∗ O O f ∗ o o Since Spec Z is tensorial (Lemma 4.4), the functor f ′∗ is algebraic. Thus, f ′∗ p ∗ ≃ r ∗ f ∗ is algebraic. Descent along r : V → T (Lemma 4.2(iii)) impliesthat f ∗ is algebraic. (cid:3) Tensor localizations
The goal of this section is to prove the following theorem.
Theorem 5.1.
Let X be a quasi-compact and quasi-separated algebraicstack. Let i : Z → X be a finitely presented closed immersion defined byan ideal sheaf I . Let j : U → X be the open complement of Z . Let T bean algebraic stack and let f ∗ : QCoh ( X ) → QCoh ( T ) be a cocontinuous ten-sor functor. Let i T : Z T → T be the closed immersion defined by the ideal I T := Im( f ∗ I → O T ) . Let j T : U T → T denote the complement of Z T . (i) There exists a cocontinuous tensor functor f ∗ U : QCoh ( U ) → QCoh ( U T ) , which is essentially unique, and a canonical isomorphism of tensorfunctors j ∗ T f ∗ ≃ f ∗ U j ∗ . (ii) For each integer n ≥ , f ∗ Z [ n ] := ( i [ n ] T ) ∗ f ∗ ( i [ n ] ) ∗ : QCoh ( Z [ n ] ) → QCoh ( Z [ n ] T ) is a cocontinuous tensor functor and there is a canonical isomor-phism of tensor functors ( i [ n ] T ) ∗ f ∗ ≃ f ∗ Z [ n ] ( i [ n ] ) ∗ . Moreover, f ∗ ( i [ n ] ) ∗ ≃ ( i [ n ] T ) ∗ f ∗ Z [ n ] .In addition, if f ∗ preserves sheaves of finite type, then the same is true of f ∗ U and f ∗ Z [ n ] for all n ≥ . Theorem 5.1 features in a key way in the proof of our main theorem(Theorem 8.4), which we prove via stratifications and formal gluings. Fromthis context, we hope that the long and technical statement of Theorem5.1 should appear to be quite natural. While Theorem 5.1(ii) follows easilyfrom the results of § ANNAKA DUALITY 15
Let C be a Grothendieck abelian category. A Serre subcategory is afull non-empty subcategory K ⊆ C closed under taking subquotients andextensions. Serre subcategories are abelian and the inclusion functor isexact. A Serre subcategory is localizing if it is also closed under small directsums in C , equivalently, it is closed under small colimits in C .If K ⊆ C is a Serre subcategory, then there is a quotient Q of C by K and an exact functor q ∗ : C → Q , which is universal for exact functorsout of C that vanish on K [Gab62, Ch. III]. Note that K is localizing ifand only if the quotient q ∗ : C → Q is a localization , that is, q ∗ admits aright adjoint q ∗ : Q → C ; it follows that Q is Grothendieck abelian, q ∗ iscocontinuous, q ∗ is fully faithful and q ∗ q ∗ ≃ id Q . This statement followsby combining the Gabriel–Popescu Theorem (e.g., [BD68, Thm. 6.25]) with[BD68, Prop. 6.21].Let C be a Grothendieck abelian tensor category and let K ⊆ C be aSerre subcategory. We say that K is a tensor ideal if K is closed undertensor products with objects in C . If K is also localizing, then we say that K is a localizing tensor ideal .If f ∗ : C → D is an exact cocontinuous tensor functor between Grothendieckabelian tensor categories, then ker( f ∗ ) is a localizing tensor ideal. Con-versely, if K ⊆ C is a localizing tensor ideal, then the quotient Q = C / K is a Grothendieck abelian tensor category, the localization q ∗ : C → Q isan exact cocontinuous tensor functor and ker( q ∗ ) = K ; in this situation, wewill refer to q ∗ as a tensor localization . Example 5.2.
Let f : X → Y be a morphism of algebraic stacks. If f isflat, then f ∗ is exact. If f is a quasi-compact flat monomorphism (e.g., aquasi-compact open immersion), then QCoh ( X ) is the quotient of QCoh ( Y )by ker( f ∗ ). This follows from the fact that the counit f ∗ f ∗ → id is anisomorphism so that f ∗ is a section of f ∗ [Gab62, Prop. III.2.5]. Definition 5.3.
Let C be a Grothendieck abelian tensor category. For M ∈ C let ϕ M : O C → H om C ( M, M ) denote the adjoint to the canonicalisomorphism O C ⊗ C M → M . Let the annihilator Ann C ( M ) of M be thekernel of ϕ M , which we consider as an ideal of O C . Example 5.4.
Let X be an algebraic stack and let F ∈
QCoh ( X ). ThenAnn QCoh ( X ) ( F ) = QC (cid:0) Ann
Mod ( X ) ( F ) (cid:1) . In particular, if F is of finite type,then Ann QCoh ( X ) ( F ) = Ann Mod ( X ) ( F ).Recall that an object c ∈ C is finitely generated if the natural map:lim −→ λ Hom C ( c, d λ ) → Hom C ( c, lim −→ λ d λ )is bijective for every direct system { d λ } λ in C with monomorphic bondingmaps. A category C is locally finitely generated if it is cocomplete (all smallcolimits exist) and has a set A of finitely generated objects such that everyobject c of C is a directed colimit of objects from A . Example 5.5.
Let X be a quasi-compact and quasi-separated algebraicstack. The finitely generated objects in QCoh ( X ) are the quasi-coherentsheaves of finite type. Thus QCoh ( X ) is locally finitely generated [Ryd16].We also require the following definition. Definition 5.6.
Let q ∗ : C → Q be a tensor localization. Then it is sup-ported if q ∗ ( O C / Ann( K )) ∼ = 0 for every finitely generated object K of C such that q ∗ ( K ) ∼ = 0.The notion of a supported tensor localization is very natural. Example 5.7. If f : X → Y is a flat monomorphism of quasi-compact andquasi-separated algebraic stacks, then the tensor localization f ∗ : QCoh ( Y ) → QCoh ( X ) of Example 5.2 is supported. Indeed, if M is a quasi-coherent O Y -module of finite type in the kernel of f ∗ , then f ∗ Ann
QCoh ( Y ) ( M ) =Ann QCoh ( X ) ( f ∗ M ) = O X .We now have our key result, which also generalizes [BC14, Lem. 3.3.6]. Theorem 5.8.
Let C be a locally finitely generated Grothendieck abeliantensor category and let q ∗ : C → Q be a supported tensor localization. Let D be a Grothendieck abelian tensor category. If f ∗ : C → D is a cocontinuoustensor functor such that f ∗ ( K ) ∼ = 0 for every finitely generated object K of C such that q ∗ ( K ) ∼ = 0 , then f ∗ factors essentially uniquely through acocontinuous tensor functor g ∗ : Q → D . If f ∗ preserves finitely generatedobjects, then so does g ∗ . Note that Theorem 5.8 is trivial if f ∗ is exact. The challenge is to usethe symmetric monoidal structure to deduce this also when f ∗ is merelyright-exact. The proof we give is a straightforward generalization of [BC14,Lem. 3.3.6]. First, we will see how Theorem 5.8 implies Theorem 5.1. Proof of Theorem 5.1.
For (ii), note that ( i [ n ] ) ∗ identifies QCoh ( Z [ n ] ) withthe category of modules over the algebra A n = O X /I n +1 . The algebra f ∗ A n is O T /I n +1 T and ( f Z [ n ] ) ∗ = ( f A n ) ∗ in the terminology of § QCoh ( X ) is locally finitely generated (Example 5.5) andthat j ∗ : QCoh ( X ) → QCoh ( U ) is a supported localization (Example 5.7).If K ∈ QCoh ( X ) is finitely generated and j ∗ K = 0, then I m K = 0 forsufficiently large m . Thus, the natural map I m ⊗ O X K → O X ⊗ O X K ∼ = K is zero. Applying j ∗ T f ∗ , the map becomes the identity since j ∗ T f ∗ ( I m ) → j ∗ T f ∗ ( O X ) = O U T is an isomorphism. It follows that j ∗ T f ∗ K = 0. We maythus apply Theorem 5.8 and deduce that j ∗ T f ∗ factors via j ∗ and a tensorfunctor f ∗ U : QCoh ( U ) → QCoh ( U T ). (cid:3) To prove Theorem 5.8 we require the following lemma.
Lemma 5.9 ([BC14, Lem. 3.3.2]) . Let f ∗ : C → D be a cocontinuous tensorfunctor. If I ⊆ O C is an O C -ideal such that f ∗ ( O C /I ) ∼ = 0 , then f ∗ ( I ) → f ∗ ( O C ) is an isomorphism.Proof. Since f ∗ is right-exact and f ∗ ( O C /I ) = 0, it follows that f ∗ ( I ) → f ∗ ( O C ) = O D is surjective. Let J = f ∗ ( I ) and let ϕ : J → O D denote thesurjection. The multiplication I ⊗ C I → I factors through I ⊗ C O C and O C ⊗ C I and gives rise to the commutative diagram J ⊗ D J id J ⊗ ϕ / / / / ϕ ⊗ id J (cid:15) (cid:15) (cid:15) (cid:15) J ⊗ D O D ∼ = (cid:15) (cid:15) O D ⊗ D J ∼ = / / J ANNAKA DUALITY 17
Let η F denote the unit of the adjunction between − ⊗ D F and H om D ( F, − ).Then we obtain the commutative diagram J η J ( J ) / / ϕ (cid:15) (cid:15) (cid:15) (cid:15) H om D ( J, J ⊗ J ) H om ( − , id J ⊗ ϕ ) / / H om ( − ,ϕ ⊗ id J ) (cid:15) (cid:15) H om D ( J, J ⊗ O D ) ∼ = (cid:15) (cid:15) O D η J ( O D ) / / H om D ( J, O D ⊗ J ) ∼ = / / H om D ( J, J ) . But the top row also factors as J η O D ( J ) / / H om D ( O D , J ⊗ O D ) H om ( ϕ, − ) / / H om D ( J, J ⊗ O D )which is injective since η O D is an isomorphism and ϕ is surjective. It followsthat J → H om D ( J, J ) is injective, hence so is ϕ : J → O D . (cid:3) Proof of Theorem 5.8. If K ∈ C , since C is locally finitely generated, it maybe written as a directed colimit K = lim −→ λ K λ , where K λ ⊆ K and K λ isfinitely generated. If K ∈ ker( q ∗ ), then q ∗ K λ ⊆ q ∗ K ∼ = 0. In particular, K := ker( q ∗ ) ⊆ ker( f ∗ ).Let 0 → K → M → N → Q → C with K, Q ∈ K . We have to prove that f ∗ ( M → N ) is an isomorphism in D .Let N be the image of M in N . By right-exactness, we have an exactsequence f ∗ ( K ) → f ∗ ( M ) → f ∗ ( N ) →
0. Since f ∗ ( K ) = 0, we have that f ∗ ( M ) = f ∗ ( N ). We may thus replace M with N and assume that K = 0and M → N is injective.Write N as the directed colimit of finitely generated subobjects N ◦ λ ⊆ N .Let N λ = M + N ◦ λ ⊆ N and I λ = Ann( N λ /M ). By definition, we have that I λ ⊗ N λ /M → N λ /M is zero; hence I λ ⊗ N λ → N λ factors through M .Note that N λ /M = N ◦ λ / ( N ◦ λ ∩ M ) is a quotient of a finitely generatedobject and a subobject of Q , so O C /I λ ∈ K since q ∗ is supported. Weconclude that f ∗ ( I λ ) → f ∗ ( O C ) is an isomorphism using Lemma 5.9. Nowconsider the commutative diagrams: I λ ⊗ M / / (cid:15) (cid:15) M (cid:15) (cid:15) I λ ⊗ N λ / / : : ✈✈✈✈✈✈✈✈✈✈ N λ and f ∗ ( M ) ∼ = / / (cid:15) (cid:15) f ∗ ( M ) (cid:15) (cid:15) f ∗ ( N λ ) ∼ = / / ssssssssss f ∗ ( N λ ) , where the right diagram is obtained by applying f ∗ to the left diagram. Itfollows that f ∗ ( M ) → f ∗ ( N λ ) is an isomorphism. Since f ∗ is cocontinuous,it follows that f ∗ ( M ) → f ∗ ( N ) = lim −→ f ∗ ( N λ ) is an isomorphism.This proves that f ∗ = g ∗ q ∗ where g ∗ = f ∗ q ∗ . It is readily verified that g ∗ is cocontinuous (it preserves small direct sums and is right-exact). If M ∈ Q is a finitely generated object, then we may find a finitely generated object N ∈ C such that M = q ∗ N . Indeed, by assumption q ∗ M is a filtered colimitof finitely generated objects. It follows that there is a finitely generatedsubobject N ⊆ q ∗ M such that q ∗ N → M is an isomorphism. If f ∗ preservesfinitely generated objects, then g ∗ M = f ∗ N is finitely generated. (cid:3) To show how powerful tensor localization is, we can quickly prove thattensoriality is local for the Zariski topology—even for stacks.
Theorem 5.10.
Let X be a quasi-compact and quasi-separated algebraicstack. Let X = S nk =1 X k be an open covering by quasi-compact open sub-stacks. If every X k is tensorial, then so is X .Proof. Let j k : X k → X denote the open immersion and let I k be an idealof finite type defining a closed substack complementary to X k [Ryd16,Prop. 8.2].Let T be an algebraic stack. First we will show that ω X ( T ) is fully faith-ful. Thus, let f, g : T → X be two morphisms and suppose that we are givena natural transformation of cocontinuous tensor functors γ : f ∗ ⇒ g ∗ . Then f ∗ ( O X /I k ) ։ g ∗ ( O X /I k ) so there is an inclusion f − ( X k ) ⊆ g − ( X k ) forevery k . Let T k = f − ( X k ), let j k,T : T k → T denote the corresponding openimmersion and let f k , g k : T k → X k denote the restrictions of f and g . Since( f k ) ∗ = j ∗ k,T f ∗ ( j k ) ∗ and ( g k ) ∗ = j ∗ k,T g ∗ ( j k ) ∗ , we obtain a natural transfor-mation γ k : f ∗ k ⇒ g ∗ k , hence a unique 2-isomorphism f k ⇒ g k . Since T = S Nk =1 T k , it follows by fppf-descent, that ω X ( T ) is faithful (Lemma 4.2(i)).As this holds for all T , we also have that ω X ( T k ∩ T k ′ ) is faithful and itfollows by fppf-descent that ω X ( T ) is full (Lemma 4.2(ii)).For essential surjectivity, let f ∗ : QCoh ( X ) → QCoh ( T ) be a cocontinuoustensor functor. The surjection O T ։ f ∗ ( O X /I k ) defines a closed subschemeand we let j k,T : T k → T denote its open complement. By Theorem 5.1(i), j ∗ k,T f ∗ factors via j ∗ k and a tensor functor f ∗ k : QCoh ( X k ) → QCoh ( T k ). Thelatter is algebraic by assumption; hence, so is j ∗ k,T f ∗ = f ∗ k j ∗ k .Finally, since O X /I ⊗ · · · ⊗ O X /I n = 0, it follows that f ∗ ( O X /I ) ⊗ · · · ⊗ f ∗ ( O X /I n ) = 0 so T = S nk =1 T k is an open covering. We conclude that f ∗ is algebraic by fppf descent (Lemma 4.2(iii)). (cid:3) Combining Theorem 5.10 with Lemma 4.4 we obtain a short proof of themain result of [BC14].
Corollary 5.11 (Brandenburg–Chirvasitu) . Every quasi-compact and quasi-separated scheme is tensorial. The Main Lemma
The main result of this section is the following technical lemma, whichproves that the tensorial property extends over nilpotent thickenings ofquasi-compact algebraic stacks with affine stabilizers having the resolutionproperty.
Lemma 6.1 (Main Lemma) . Let i : X → X be a closed immersion ofalgebraic stacks defined by a quasi-coherent ideal I such that I n = 0 forsome integer n > . Suppose that X is quasi-compact and quasi-separatedwith affine stabilizers. If X has the resolution property, then X is tensorial. We have another lemma that will be crucial for proving Lemma 6.1.
Lemma 6.2.
Consider a -cocartesian diagram of algebraic stacks: U p (cid:15) (cid:15) (cid:31) (cid:127) i / / U p (cid:15) (cid:15) X (cid:31) (cid:127) j / / X, ANNAKA DUALITY 19 such that the following conditions are satisfied. (i) i is a nilpotent closed immersion; (ii) U is an affine scheme; and (iii) X is quasi-compact and quasi-separated with affine stabilizers.If X has the resolution property, then so has X .Proof. Note that X has affine diagonal by the Totaro–Gross theorem; hence p is affine. By [Hal14b, Prop. A.2], the square is a geometric pushout. Inparticular, j is a nilpotent closed immersion, p is affine, and the naturalmap O X → p ∗ O U × p ∗ i ∗ O U j ∗ O X is an isomorphism. By the Totaro–GrossTheorem [Gro13, Cor. 5.9], there exists a vector bundle V on X such thatthe total space of the frame bundle of V is quasi-affine. Let E = p ∗ V ;then, since U is affine, there exists a vector bundle E on U equipped withan isomorphism α : i ∗ E → E . Let V be the quasi-coherent O X -module p ∗ E × α j ∗ V . By [Fer03, Thm. 2.2(iv)], V is a vector bundle on X and thereis an isomorphism j ∗ V ∼ = V . By [Gro13, Prop. 5.7], it follows that X hasthe resolution property. (cid:3) Proof of Lemma 6.1.
We prove the result by induction on n >
0. The case n = 1 is Theorem 4.10. So we let n > W ֒ → W is any closed immersion of algebraic stacks defined by anideal J such that J n − = 0 and W has the resolution property, then W is tensorial. We now fix a closed immersion of algebraic stacks i : X → X defined by an ideal I such that I n = 0 and X has the resolution property.It remains to prove that X is tensorial.We observe that the Totaro–Gross Theorem [Gro13, Cor. 5.9] impliesthat X has affine diagonal; thus, X has affine diagonal. We have seen that ω X ( T ) is fully faithful (Proposition 4.8) so it remains to prove that ω X ( T )is essentially surjective. By descent, it suffices to prove that if T is an affinescheme and f ∗ : QCoh ( X ) → QCoh ( T ) is a cocontinuous tensor functor, thenthere exists an ´etale and surjective morphism c : T ′ → T such that c ∗ f ∗ isalgebraic (Lemma 4.2(iii)).By Corollary 3.6, there is a 2-cocartesian diagram in GTC
QCoh ( T ) QCoh ( X ) f ∗ o o QCoh ( T ) k ∗ O O QCoh ( X ) , i ∗ O O f ∗ o o where k : T → T is the closed immersion defined by the image K of f ∗ I in O T . In particular, K n = 0. Since X has the resolution property, f ∗ isgiven by a morphism of algebraic stacks f : T → X (Theorem 4.10).Let p : U → X be a smooth and surjective morphism, where U is anaffine scheme; then, p is affine. The pullback of p along the morphism i ◦ f : T → X results in a smooth and affine surjective morphism of schemes q : V → T . By [EGA, IV.17.16.3(ii)], there exists an affine ´etale andsurjective morphism c : T ′ → T such that the pullback q ′ : V ′ → T ′ of q to T ′ admits a section. By [EGA, IV.18.1.2], there exists a unique affine´etale morphism c : T ′ → T lifting c : T ′ → T . After replacing T with T ′ and f ∗ with c ∗ f ∗ , we may thus assume that q admits a section (Lemma4.2(iii)).Let X ′ = S pec X ( f ∗ O T ). Let I ′ = I ( f ∗ O T ) be the O X ′ -ideal generated by I and let X ′ = V ( I ′ ). Then X ′ is a quasi-compact stack with affine diagonal, X ′ → X ′ is a closed immersion defined by an ideal whose n th power vanishesand X ′ has the resolution property. Let f ′∗ = ¯ f ∗ : QCoh ( X ′ ) → QCoh ( T )be the resulting tensor functor.Since f ′∗ is right-exact, it follows that K = im( f ′∗ I ′ → O T ). Also, I ′ ⊆ f ′∗ K ⊆ O X ′ . Thus V ( f ′∗ K ) ⊆ X ′ , so has the resolution property.Note that f ′∗ I ′ → f ′∗ f ′∗ K → K is surjective. Since f ′∗ is lax symmetricmonoidal, for each integer l ≥ f ′∗ K ) ⊗ l → O X ′ factorsthrough f ′∗ ( K ⊗ l ) → O X ′ . In particular, ( f ′∗ ( K l )) ⊆ f ′∗ ( K l +1 ) and ( f ′∗ K ) n =0. We may thus replace X by X ′ , X by V ( f ′∗ K ), f ∗ by f ′∗ , I by f ∗ K andassume henceforth that(i) O X → f ∗ O T is an isomorphism,(ii) I = f ∗ K for some O T -ideal K with K n = 0,(iii) f ∗ ( K l ) ⊆ f ∗ ( K l +1 ) for each integer l ≥ f ∗ K ) l ⊆ f ∗ ( K l ) for l ≥
1, and(v) q : V → T admits a section.For each integer l ≥ I l = f ∗ ( K l +1 ), which is a quasi-coherent sheaf ofideals on X . Let i l : X l → X be the closed immersion defined by I l and let k l : T l → T be the closed immersion defined by K l +1 . Since f ∗ f ∗ ( K l +1 ) → f ∗ O X = O T factors through K l +1 , it follows that k ∗ l f ∗ ( i l ) ∗ ( O X l ) = O T l .Hence, f ∗ l = k ∗ l f ∗ ( i l ) ∗ : QCoh ( X l ) → QCoh ( T l ) is a tensor functor and k ∗ l f ∗ ≃ f ∗ l ( i l ) ∗ (Theorem 5.1(ii)).By condition (iv), we see that i l : X → X l is a closed immersion of alge-braic stacks defined by an ideal whose ( l + 1)th power is zero. In particular,if l < n −
1, then X l is tensorial by the inductive hypothesis. Thus, thetensor functor f ∗ l is given by an affine morphism f l : T l → X l .We will now prove by induction on l ≥ X l has the resolutionproperty. Since X n − = X , the result will then follow from Theorem 4.10.Note that (iii) implies that the closed immersion X l → X l +1 is a square zeroextension of X l by I l /I l +1 . Let m = n − Claim 1. If M ∈ QCoh ( T m ), the natural map f ∗ ( k m ) ∗ M → p ∗ p ∗ f ∗ ( k m ) ∗ M is split injective. Proof of Claim 1.
Form the cartesian diagram of algebraic stacks: V q (cid:15) (cid:15) / / V mq m (cid:15) (cid:15) g m / / U mp m (cid:15) (cid:15) u m / / U p (cid:15) (cid:15) T / / T m f m / / X m i m / / X. Now observe that f ∗ ( k m ) ∗ M ∼ = ( i m ) ∗ ( f m ) ∗ M . Since f ∗ m is given by a mor-phism f m : T m → X m , there are natural isomorphisms: p ∗ p ∗ f ∗ ( k m ) ∗ M ∼ = p ∗ p ∗ ( i m ) ∗ ( f m ) ∗ M ∼ = p ∗ ( u m ) ∗ p ∗ m ( f m ) ∗ M ∼ = p ∗ ( u m ) ∗ ( g m ) ∗ q ∗ m M ∼ = ( i m ) ∗ ( f m ) ∗ ( q m ) ∗ q ∗ m M. ANNAKA DUALITY 21
Hence, it remains to prove that the natural map M → ( q m ) ∗ q ∗ m M is splitinjective. But q m is affine, so ( q m ) ∗ q ∗ m M ∼ = ( q m ) ∗ O V m ⊗ O Tm M . Thus,we are reduced to proving that O T m → ( q m ) ∗ O V m is split injective. By(v), q admits a section. Since q m is smooth and T m is affine, the sectionthat q admits lifts to a section of q m . This implies that the morphism O T m → ( q m ) ∗ O V m is split injective. △ Claim 2.
If 0 ≤ l < n −
1, then the natural maps I l /I l +1 → p ∗ p ∗ ( I l /I l +1 )are split injective. Proof of Claim 2. If N ∈ QCoh ( T m ), then f ∗ ( k m ) ∗ N = ( i m ) ∗ ( f m ) ∗ N . Since f m is an affine morphism, it follows that f ∗ ( k m ) ∗ : QCoh ( T m ) → QCoh ( X )is exact. If P is one of the modules K l +1 , K l +2 , or K l +1 /K l +2 , then K m +1 P = 0, so the natural map P → ( k m ) ∗ k ∗ m P is an isomorphism. Inparticular, f ∗ P ∼ = f ∗ ( k m ) ∗ k ∗ m P . Hence, I l /I l +1 ∼ = f ∗ ( k m ) ∗ k ∗ m ( K l +1 /K l +2 )and the claim now follows from Claim 1. △ So we let l ≥ < n −
1. We willassume that X l has the resolution property and we will now prove that X l +1 has the resolution property. Retaining the notation of Claim 1, there is a2-commutative diagram of algebraic stacks: U lp l (cid:15) (cid:15) / / U l +1 (cid:15) (cid:15) ˜ X l +1 (cid:15) (cid:15) X l / / ❧❧❧❧❧❧❧❧❧❧❧❧❧ X l +1 , where both the inner and outer squares are 2-cartesian and the inner squareis 2-cocartesian. Let Q l = I l /I l +1 . The morphism X l → ˜ X l +1 is a squarezero extension of X l by ( p l ) ∗ p ∗ l Q l ∼ = p ∗ p ∗ Q l and the morphism ˜ X l +1 → X l +1 is the morphism of X l -extensions given by the natural map Q l → ( p l ) ∗ p ∗ l Q l .By Claim 2, the morphism Q l → ( p l ) ∗ p ∗ l Q l is split injective and so there isan induced splitting X l +1 → ˜ X l +1 which is affine. By [Gro13, Prop. 4.3(i)],it remains to prove that ˜ X l +1 has the resolution property, which is justLemma 6.2. (cid:3) Formal gluings
Let T be an algebraic stack, let i : Z ֒ → T be a finitely presented closedimmersion and let j : U → T denote its complement. A flat Mayer–Vietorissquare is a cartesian square of algebraic stacks U ′ j ′ / / π U (cid:15) (cid:15) T ′ π (cid:15) (cid:15) U j / / T (cid:3) such that π is flat and π | Z is an isomorphism [MB96, HR16]. If F : AlgSt op → Cat is a pseudo-functor, then there is a natural functor:Φ F : F ( T ) → F ( T ′ ) × F ( U ′ ) F ( U ) . Here
AlgSt denotes the 2-category of algebraic stacks. For the purposesof this paper, it is enough to consider pseudo-functors defined on affineschemes, that is, fibered categories over affine schemes. Indeed, our Mayer–Vietoris squares will be formal gluings : T = Spec A is affine and noetherian, Z = V ( I ), and T ′ = Spec b A , where b A is the I -adic completion of A .The following theorem follows from the main results of [HR16] (and almostfrom [MB96]). Theorem 7.1.
Consider a flat Mayer–Vietoris square as above. Let X be analgebraic stack and consider the pseudo-functor X ⊗ ( − ) = Hom ⊗ ( QCoh ( X ) , QCoh ( − )) on the category of algebraic spaces. (i) Φ X ⊗ is an equivalence of categories; (ii) Φ X is fully faithful; (iii) Φ X is an equivalence if ∆ X is quasi-affine; (iv) Φ X is an equivalence if X is Deligne–Mumford; and (v) Φ X is an equivalence if T is locally excellent.Proof. By [HR16, Thm. B(1)] (or one of [MB96, 0.3] and [FR70, App.] when π is affine), there is an equivalence QCoh ( T ) → QCoh ( T ′ ) × QCoh ( U ′ ) QCoh ( U ) . Thus we have (i). Claims (ii) and (iii) are [HR16, Thm. B(3)] and claims(iv) and (v) are [HR16, Thm. E and Thm. A] respectively. Under someadditional assumptions: π is affine, ∆ X is quasi-compact and separated,and in (v) T ′ is locally noetherian; claims (ii)–(v) also follow from [MB96,6.2 and 6.5.1]. (cid:3) Remark . Recall that a noetherian ring A is excellent [Mat89, p. 260],[Mat80, Ch. 13] or [EGA, IV.7.8.2], if(i) A is a G-ring, that is, A p → c A p has geometrically regular fibers;(ii) the regular locus Reg B ⊆ Spec B is open for every finitely generated A -algebra B ; and(iii) A is universally catenary.If (i) and (ii) hold, then we say that A is quasi-excellent . All excellencyassumptions originate from [MB96, HR16] via Theorem 7.1. The assump-tions are used to guarantee that the formal fibers are geometrically regularso that N´eron–Popescu desingularization applies. We can thus replace “lo-cally excellent” with “locally the spectrum of a G-ring”. Note that whereasbeing a G-ring and being quasi-excellent are local for the smooth topol-ogy [Mat89, 32.2], excellency does not descend even for finite ´etale cover-ings [EGA, IV.18.7.7]. Corollary 7.3.
Let X be an algebraic stack. Let A be a ring and let I ⊂ A be a finitely generated ideal. Let T = Spec A , Z = V ( I ) and U = T \ Z . Let i : Z → T and j : U → T be the resulting immersions. (i) Let f , f : T → X be morphisms of algebraic stacks. ANNAKA DUALITY 23 (a)
Assume that ker( O T → j ∗ O U ) ∩ T ∞ n =0 I n = 0 . Let α , β : f ⇒ f be -morphisms. If α U = β U and α Z [ n ] = β Z [ n ] for all n ,then α = β . (b) Assume that T is noetherian and that ω X ( T ) is faithful for allnoetherian T . Let t : f ∗ ⇒ f ∗ be a natural transformation ofcocontinuous tensor functors. If j ∗ ( t ) and ( i [ n ] ) ∗ ( t ) are realiz-able for all n , then t is realizable. (ii) Assume either (a) T is excellent, or (b) T is noetherian and X hasquasi-affine or unramified diagonal. Further, assume that ω X, ≃ ( T ) is fully faithful for all noetherian T . Let f ∗ : QCoh ( X ) → QCoh ( T ) be a cocontinuous tensor functor that preserves sheaves of finitetype . If j ∗ f ∗ and ( i [ n ] ) ∗ f ∗ are algebraic for all n , then f ∗ is alge-braic. The assumption in (i)(a) says that the filtration {∅ ֒ → Z ֒ → T } is sepa-rating (Definition A.1). This is automatic if T is noetherian (Lemma A.2). Proof of Corollary 7.3.
First, we show (ii). By assumption, the inducedfunctor ( i [ n ] ) ∗ f ∗ comes from a morphism f [ n ] : Z [ n ] → X . Pick an ´etalecover q : ˜ Z → Z such that f [0] ◦ q : ˜ Z → X has a lift g : ˜ Z → W , where p : W → X is a smooth covering and W is affine. Descent (Lemma 4.2(iii))implies that we are free to replace T with an ´etale cover, so we may assumethat f also has a lift g : Z → W [EGA, IV.18.1.1].Since p is smooth, we may choose compatible lifts g [ n ] : Z [ n ] → W of f [ n ] for all n . But W is affine, so there is an induced morphism ˆ g : ˆ T → W , whereˆ T = Spec ˆ A and ˆ A denotes the completion of A at the ideal I . Let ˆ f = p ◦ ˆ g .Then ( i [ n ] ) ∗ ˆ f ∗ = ( f [ n ] ) ∗ = ( i [ n ] ) ∗ f ∗ for all n . Since Coh ( ˆ T ) = lim ←− n Coh ( Z n )(Lemma 3.8), it follows that ˆ f ∗ ≃ π ∗ f ∗ where π : ˆ T → T is the completionmorphism. Indeed, this last equivalence may be verified after restrictingboth sides to quasi-coherent O X -modules of finite type (Example 5.5) andboth sides send quasi-coherent O X -modules of finite type to Coh ( ˆ T ).Let ˆ : ˆ U → ˆ T be the pullback of j along π ; then we obtain a flat Mayer–Vietoris square: ˆ U ′ ˆ / / π U (cid:15) (cid:15) ˆ T π (cid:15) (cid:15) U j / / T. (cid:3) Since U and ˆ U are noetherian, ω X, ≃ ( U ) and ω X, ≃ ( ˆ U ) are fully faithful.Thus, there is an essentially unique morphism of algebraic stacks h : U → X such that h ∗ ≃ j ∗ f ∗ . But there are isomorphisms:ˆ ∗ ˆ f ∗ ≃ ˆ ∗ π ∗ f ∗ ≃ π ∗ U j ∗ f ∗ ≃ π ∗ U h ∗ , so ˆ f ◦ ˆ ≃ h ◦ π U . That f ∗ is algebraic now follows from Theorem 7.1.For (i)(b), we proceed similarly. Consider the representable morphism E → T given by the equalizer of f and f . Then 2-isomorphisms between f and f correspond to T -sections of E . By assumption, we have compatiblesections τ U ∈ E ( U ) and τ [ n ] ∈ E ( Z [ n ] ) for all n . Choose an ´etale presentation E ′ → E by an affine scheme E ′ . We may replace T with an ´etale cover (Lemma 4.2(ii)) and thus assume that τ [0] lifts to E ′ . In particular, thereare compatible lifts of all the τ [ n ] to E ′ . Since E ′ is affine, we get an inducedmorphism ˆ T → E ′ ; thus, a morphism ˆ T → E . Equivalently, we get a 2-isomorphism between f ◦ π and f ◦ π . The induced 2-isomorphism between π ∗ f ∗ and π ∗ f ∗ equals π ∗ t since it coincides on the truncations. We may nowapply Theorem 7.1 to deduce that t is realized by a 2-morphism τ : f ⇒ f .For (i)(a), we consider the representable morphism r : R → T given bythe equalizer of α and β . It suffices to prove that r is an isomorphism.Note that r is always a monomorphism and locally of finite presentation.By assumption, there are compatible sections of r over U and Z [ n ] for all n ,thus r U and r Z [ n ] are isomorphisms for all n . By Proposition A.3, r is anisomorphism. (cid:3) Remark . We do not know if the condition that f ∗ preserves sheaves offinite type in (ii) is necessary. We do know that for any sheaf F of finitetype, the restrictions of f ∗ F to U and Z [ n ] are coherent but this does notimply that f ∗ F is coherent. For example, if A = k [[ x ]], and I = ( x ), thenthe A -module k (( x )) /k [[ x ]] is not finitely generated but becomes 0 aftertensoring with A/ ( x n ) or A x .8. Tannaka duality
In this section, we prove our general Tannaka duality result (Theorem 8.4)and as a consequence also establish Theorem 1.1. To accomplish this, weconsider the following refinement of [HR15a, Def. 2.5].
Definition 8.1.
Let X be a quasi-compact algebraic stack. A finitely pre-sented filtration of X is a sequence of finitely presented closed immersions ∅ = X ֒ → X ֒ → X ֒ → . . . ֒ → X r ֒ → X such that | X r | = | X | . The strata of the filtration are the locally closed finitely presented substacks Y k := X k \ X k − . The n th infinitesimal neighborhood of X k is the finitelypresented closed immersion X [ n ] k ֒ → X which is given by the ideal I n +1 k where X k ֒ → X is given by I k . The n th infinitesimal neighborhood of Y k isthe locally closed finitely presented substack Y [ n ] k := X [ n ] k \ X k − .Stacks that have affine stabilizers can be stratified into stacks with theresolution property. Proposition 8.2.
Let X be an algebraic stack. The following are equivalent: (i) X is quasi-compact and quasi-separated with affine stabilizers; (ii) X has a finitely presented filtration ( X k ) with strata of the form Y k = [ U k / GL N k ] where U k is quasi-affine. (iii) X has a finitely presented filtration ( X k ) with strata Y k that arequasi-compact with affine diagonal and the resolution property.Proof. That (i) = ⇒ (ii) is [HR15a, Prop. 2.6(i)]. That (ii) ⇐⇒ (iii) is theTotaro–Gross theorem [Gro13]. That (iii) = ⇒ (i) is straightforward. (cid:3) When in addition X is noetherian or, more generally, X has finitely pre-sented inertia, this result is due to Kresch [Kre99, Prop. 3.5.9] and Drinfeld–Gaitsgory [DG13, Prop. 2.3.4]. They construct stratifications by quotientstacks of the form [ V k / GL N k ], where each V k is quasi-projective and the ANNAKA DUALITY 25 action is linear. This implies that the strata have the resolution property.When X has finitely presented inertia the situation is simpler since X canbe stratified into gerbes [Ryd16, Cor. 8.4], something which is not possiblein general. Remark . In [DG13, Def. 1.1.7], Drinfeld and Gaitsgory introduces thenotion of a QCA stack. These are (derived) algebraic stacks that are quasi-compact and quasi-separated with affine stabilizers and finitely presentedinertia. The condition on the inertia is presumably only used for [DG13,Prop. 2.3.4] and could be excised using Proposition 8.2.We now state and prove the main result of the paper.
Theorem 8.4.
Let T and X be algebraic stacks and consider the functor ω X ( T ) : Hom( T, X ) → Hom c ⊗ ( QCoh ( X ) , QCoh ( T )) and its variants ω ft X ( T ) , ω X, ≃ ( T ) and ω ft X, ≃ ( T ) (see § X isquasi-compact and quasi-separated. (i) If X has quasi-affine diagonal, then (a) ω X ( T ) is fully faithful; and (b) ω ft X ( T ) is essentially surjective if T is locally noetherian. (ii) If X has affine stabilizers, then (a) ω X ( T ) is faithful if T is locally noetherian or has no embeddedcomponents; (b) ω X, ≃ ( T ) is full if T is locally noetherian; (c) ω X ( T ) is full if X is affine-pointed and T is locally noetherian. (d) ω ft X ( T ) is essentially surjective if T is locally excellent, or T islocally noetherian and X is Deligne–Mumford.In particular, ω ft X, ≃ ( T ) is an equivalence if X has affine stabilizers and T islocally excellent, and ω ft X ( T ) is an equivalence if T is locally noetherian and X either has quasi-affine diagonal or is Deligne–Mumford.Proof. When X has quasi-affine diagonal, we have already seen that ω X ( T )is fully faithful for all T (Proposition 4.8). This is (i)(a).Choose a filtration ( X k ) with strata ( Y k ) as in Proposition 8.2. We willprove the theorem by induction on the number of strata r . If r = 0, then X = ∅ and there is nothing to prove. If r ≥
1, then U := X \ X has afiltration of length r −
1; thus by induction the theorem holds for U . Thetheorem also holds for X [ n ]1 = Y [ n ]1 and all n , since ω X [ n ]1 ( T ) is an equivalenceof categories by the Main Lemma 6.1. Note that if r = 1, then U = ∅ and X = X [ n ]1 = Y [ n ]1 for sufficiently large n .Let I ⊆ O X be the ideal defining Z = X . Let i [ n ] : Z [ n ] ֒ → X be theclosed substack defined by I n +1 and let j : U → X be its complement.For (ii)(a), pick two maps f , f : T → X and 2-isomorphisms τ , τ : f ⇒ f and assume that ω X ( T )( τ ) = ω X ( T )( τ ). We need to prove that τ = τ .For (ii)(b) (resp. (ii)(c)), pick two maps f , f : T → X and a natural iso-morphism (resp. transformation) γ : f ∗ ⇒ f ∗ of cocontinuous tensor func-tors. We need to prove that γ is realizable. For (i)(b) and (ii)(d), pick a cocontinuous tensor functor f ∗ : QCoh ( X ) → QCoh ( T ) preserving sheaves of finite type. We need to prove that f ∗ isalgebraic.When we prove (ii)(d) (resp. (ii)(b) and (ii)(c)), we assume that (ii)(b)(resp. (ii)(a)) already has been established. When we prove (i)(b), we notethat ω X ( T ) is fully faithful for all T . By Lemma 4.2, it is enough to provethe results when T = Spec A is affine.In cases (ii)(a), (ii)(b) and (ii)(c), let I T = Im( f ∗ I → f ∗ O X = O T ),which is a finitely generated ideal because f is a morphism. In cases (i)(b)and (ii)(d), let I T = Im( f ∗ I → f ∗ O X = O T ), which is a finitely generatedideal because T is noetherian. Let i [ n ] T : Z [ n ] T ֒ → T be the finitely presentedclosed immersion defined by I n +1 T and let j T : U T ֒ → T be its complement,a quasi-compact open immersion.In cases (ii)(a), (ii)(b) and (ii)(c), we have that U T = f − ( U ) = f − ( U );in the first case this is obvious and for the other two cases this followsfrom Corollary 4.9 and Lemma 4.7, respectively. We also have that Z [ n ] T = f − ( Z [ n ] ) ֒ → f − ( Z [ n ] ). Thus, after restricting to either Z [ n ] T or U T we havethat τ = τ in case (ii)(a) and that γ is realizable in cases (ii)(b) and (ii)(c).In cases (i)(b) and (ii)(d), Theorem 5.1 produces for every n ≥ f ∗ U : QCoh ( U ) → QCoh ( U T ) and f ∗ Z [ n ] : QCoh ( Z [ n ] ) → QCoh ( Z [ n ] T ) such that j ∗ T f ∗ ≃ f ∗ U j ∗ and ( i [ n ] T ) ∗ f ∗ ≃ ( f Z [ n ] ) ∗ ( i [ n ] ) ∗ . By the inductive assumption, f ∗ U is algebraic and the case r = 1 implies that f ∗ Z [ n ] is algebraic for each n ≥
0. In particular, j ∗ T f ∗ and( i [ n ] T ) ∗ f ∗ is algebraic for each n ≥ T is noetherian or has no embedded associated points, then the strat-ification ∅ ⊂ Z T ⊂ T is separating by Lemma A.2. The result now followsfrom Corollary 7.3. (cid:3) Remark . Let X be a quasi-compact and quasi-separated algebraic stackwith affine stabilizers. Let T be a locally noetherian stack and let π : T ′ → T be a flat morphism. Assume that we have morphisms f , f : T → X .Then Hom( f ◦ π, f ◦ π ) → Hom ⊗ ( π ∗ f ∗ , π ∗ f ∗ ) is injective even if T ′ isnot noetherian. Indeed, the stratification on T ′ constructed in the proof ofTheorem 8.4 (ii)(a), is the pull-back along π of a stratification on T , henceseparating by Lemma A.2.We conclude with the proof of Theorem 1.1. Proof of Theorem 1.1.
We note thatHom r ⊗ , ≃ ( Coh ( X ) , Coh ( T )) → Hom ft c ⊗ , ≃ ( QCoh ( X ) , QCoh ( T ))is an equivalence of categories. It is thus enough to prove that ω ft X, ≃ ( T ) isan equivalence of groupoids, which follows from Theorem 8.4. (cid:3) Applications
In this section, we address the applications outlined in the introduction.
ANNAKA DUALITY 27
Proof of Corollary 1.4.
Let T ′ → T be an fpqc covering with T locally ex-cellent and T ′ locally noetherian. Since X is an fppf-stack, we may as-sume that T and T ′ are affine and that T ′ → T is faithfully flat. Let T ′′ = T ′ × T T ′ . Since X has affine stabilizers, the functor ω X, ≃ ( T ) is anequivalence, the functor ω X, ≃ ( T ′ ) is fully faithful and the functor ω X ( T ′′ ) isfaithful for morphisms T ′′ → T ′ → X (Theorem 8.4 and Remark 8.5). SinceHom c ⊗ , ≃ ( QCoh ( X ) , QCoh ( − )) is an fpqc stack, it follows that T ′ → T is amorphism of effective descent for X . (cid:3) Proof of Corollary 1.5.
It is readily verified that we can assume that X isquasi-compact. As A is noetherian, Coh ( A ) = lim ←− n Coh ( A/I n ). Thus, X ( A ) ∼ = Hom r ⊗ , ≃ ( Coh ( X ) , Coh ( A )) ∼ = Hom r ⊗ , ≃ ( Coh ( X ) , lim ←− Coh ( A/I n )) ∼ = lim ←− Hom r ⊗ , ≃ ( Coh ( X ) , Coh ( A/I n )) ∼ = lim ←− X ( A/I n ) . (cid:3) Proof of Theorem 1.2.
First, we prove (i). We begin with the following stan-dard reductions: we can assume that S is affine; X → S is quasi-compact,so is of finite presentation; and S is of finite type over Spec Z .Since S is now assumed to be excellent, we can prove the algebraic-ity of Hom S ( Z, X ) using a variant of Artin’s criterion for algebraicity dueto the first author [Hal14b, Thm. A]. Hence, it is sufficient to prove thatHom S ( Z, X ) is[1] a stack for the ´etale topology;[2] limit preserving, equivalently, locally of finite presentation;[3] homogeneous, that is, satisfies a strong version of the Schlessinger–Rim criteria;[4] effective, that is, formal deformations can be algebraized;[5] the automorphisms, deformations, and obstruction functors are co-herent.The main result of this article provides a method to prove [4] in maximumgenerality, which we address first. Thus, let T = Spec B → S , where ( B, m )is a complete local noetherian ring. Let T n = Spec( B/ m n +1 ). Since Z → S is proper, for every noetherian algebraic stack W with affine stabilizers thereare equivalencesHom( Z × S T, W ) ∼ = Hom r ⊗ , ≃ ( Coh ( W ) , Coh ( Z × S T )) (Theorem 1.1) ∼ = Hom r ⊗ , ≃ ( Coh ( W ) , lim ←− Coh ( Z × S T n )) [Ols05, Thm. 1.4] ∼ = lim ←− Hom r ⊗ , ≃ ( Coh ( W ) , Coh ( Z × S T n )) (Lemma 3.8) ∼ = lim ←− Hom( Z × S T n , W ) (Theorem 1.1) . Since X and S have affine stabilizers, it follows thatHom S ( Z × S T, X ) ∼ = lim ←− Hom S ( Z × S T n , X );that is, the stack Hom S ( Z, X ) is effective and so satisfies [4].The remainder of Artin’s conditions are routine, so we will just sketch thearguments and provide pointers to the literature where they are addressed inmore detail. Condition [1] is just ´etale descent and [2] is standard—see, for example, [LMB, Prop. 4.18]. For conditions [3] and [5], it will be convenientto view Hom S ( Z, X ) as a substack of another moduli problem. This lets usavoid having to directly discuss the deformation theory of non-representablemorphisms of algebraic stacks.If W → S is a morphism of algebraic stacks, let Rep W/S denote the S -groupoid that assigns to each S -scheme T the category of representable morphisms of algebraic stacks V → W × S T such that the composition V → W × S T → T is proper, flat and of finite presentation. There is a mor-phism of S -groupoids: Γ : Hom S ( Z, X ) → Rep Z × S X/S , which is given bysending a T -morphism f : Z × S T → X × S T to its graph Γ( f ) : Z × S T → ( Z × S X ) × S T . It is readily seen that Γ is formally ´etale since Z → S is flat.Hence, it is sufficient to verify conditions [3] and [5] for Rep Z × S X/S [Hal14b,Lemmas 1.5(9), 6.3 & 6.11]. That Rep Z × S X/S is homogeneous follows im-mediately from [Hal14b, Lem. 9.3]. A description of the automorphism,deformation and obstruction functors of Rep Z × S X/S in terms of the cotan-gent complex are given on [Hal14b, p. 37], which mostly follows from theresults of [Ols06a]. That these functors are coherent is [Hal14a, Thm. C].This completes the proof of (i).We now address (ii) and (iii), that is, the separation properties of thealgebraic stack Hom S ( Z, X ) relative to S . Let T be an affine scheme. Let Z T and X T denote Z × S T and X × S T , respectively. Suppose we are giventwo T -morphisms f , f : Z T → X T and consider Q := Isom Z T ( f , f ) = X × X × S X Z T . Then Q → Z T is representable and of finite presentation.If π : Z T → T denotes the structure morphism, then π ∗ Q is an algebraicspace which is locally of finite presentation, being the pull-back of the diag-onal of Hom S ( Z, X ) along the morphism T → Hom S ( Z, X ) × S Hom S ( Z, X )corresponding to ( f , f ).Let P be one of the properties: affine, quasi-affine, separated, quasi-separated. Assume that ∆ X has P ; then Q → Z T has P . We claimthat the induced morphism π ∗ Q → T has P . For the properties affineand quasi-affine, this is [HR15b, Thm. 2.3 (i),(ii)]. For quasi-separated(resp. separated), this is [HR15b, Thm. 2.3 (ii),(iv)] applied to the quasi-affine morphism (resp. closed immersion) Q → Q × Z Q and the Weil re-striction π ∗ Q → π ∗ Q × T π ∗ Q = π ∗ ( Q × Z Q ). In particular, we have provedthat Hom S ( Z, X ) is algebraic and locally of finite presentation with quasi-separated diagonal over S .Now by Theorem B.1, ∆ Hom S ( Z,X ) /S = Hom S ( Z, ∆ X/S ) is of finite pre-sentation, so Hom S ( Z, X ) is also quasi-separated. It remains to prove thatit has affine stabilizers. To see this, we may assume that T is the spectrumof an algebraically closed field. In this situation, either π ∗ Q is empty or f ≃ f ; it suffices to treat the latter case. In the latter case, T → X × S X factors through the diagonal ∆ X/S : X → X × S X , so it is sufficient to provethat Hom S ( Z, I
X/S ), where I X/S : X × X × S X X → X is the inertia stack, hasaffine fibers. But I X/S defines a group over X with affine fibers, and theresult follows from Theorem B.1. (cid:3) ANNAKA DUALITY 29
Lemma 9.1.
Let f : Z → S be a proper and flat morphism of finite pre-sentation between algebraic stacks. For any morphism X → Z of algebraicstacks, the forgetful morphism f ∗ X → Hom S ( Z, X ) is an open immersion.Proof. It is sufficient to prove that if T is an affine S -scheme and h : Z × S T → X × S T is a T -morphism, then the locus of points where f T ◦ h : Z × S T → Z × S T is an isomorphism is open on T .First, consider the diagonal of f T ◦ h . This morphism is proper andrepresentable and the locus on T where this map is a closed immersion isopen [Ryd11b, Lem. 1.8 (iii)]. We may thus assume that f T ◦ h is repre-sentable. Repeating the argument on f T ◦ h , we may assume that f T ◦ h isa closed immersion. That the locus in T where f T ◦ h is an isomorphism isopen now follows easily by studying the ´etale locus of f T ◦ h , cf. [Ols06b,Lem. 5.2]. The result follows. (cid:3) Proof of Theorem 1.3.
That f ∗ X → S is algebraic, locally of finite presenta-tion, with quasi-compact and quasi-separated diagonal and affine stabilizersfollows from Theorem 1.2 and Lemma 9.1. The additional separation prop-erties of f ∗ X follows from [HR15b, Thm. 2.3 (i), (ii) & (iv)] applied to thediagonal and double diagonal of X → Z . (cid:3) As claimed in the introduction, we now extend [HR15b, Thm. 2.3 &Cor. 2.4]. The statement of the following corollary uses the notion of amorphism of algebraic stacks that is locally of approximation type [HR15b, § Corollary 9.2.
Let f : Z → S be a proper and flat morphism of finitepresentation between algebraic stacks. (i) Let h : X → S be a morphism of algebraic stacks with affine sta-bilizers that is locally of approximation type. Then Hom S ( Z, X ) isalgebraic and locally of approximation type with affine stabilizers.If h is locally of finite presentation, then so is Hom S ( Z, X ) → S . Ifthe diagonal of h is affine (resp. quasi-affine, resp. separated), thenso is the diagonal of Hom S ( Z, X ) → S . (ii) Let g : X → Z be a morphism of algebraic stacks such that f ◦ g : X → S has affine stabilizers and is locally of approximation type.Then the S -stack f ∗ X is algebraic and locally of approximation typewith affine stabilizers. If g is locally of finite presentation, thenso is f ∗ X → S . If the diagonal of g is affine (resp. quasi-affine,resp. separated), then so is the diagonal of f ∗ X → S .Proof. For (i), we may immediately reduce to the situation where S is anaffine scheme. Since f is quasi-compact, we may further assume that h isquasi-compact. By [HR15b, Lem. 1.1], there is an fppf covering { S i → S } such that each S i is affine and X × S S i → S i factors as X × S S i → X i → S i , where X i → S i is of finite presentation and X × S S i → X i is affine.Combining the results of [HR15a, Thm. 2.8] with [Ryd15, Thms. D & 7.10],we can arrange so that each X i → S has affine stabilizers (or has one of theother desired separation properties).Thus, we may now replace S by S i and may assume that X → S fac-tors as X q −→ X → S , where q is affine and X → S is of finite pre-sentation with the appropriate separation condition. By Theorem 1.2, thestack Hom S ( Z, X ) is algebraic and locally of finite presentation with theappropriate separation condition. By [HR15b, Thm. 2.3(i)], the morphismHom S ( Z, X ) → Hom S ( Z, X ) is representable by affine morphisms; the re-sult follows.For (ii) we argue exactly as in the proof of Theorem 1.3. (cid:3) Counterexamples
In this section we give four counter-examples (Theorems 10.1, 10.2, 10.4,and 10.5): • in Theorems 1.1, 1.2 and 8.4(ii)(a) it is necessary that X has affinestabilizer groups; • in Theorem 8.4(ii)(c), it is necessary that X is affine-pointed; • in Theorem 1.2, it is necessary that X has affine stabilizer groups;and • in Corollary 1.5, it is necessary that X has affine stabilizer groups. Theorem 10.1.
Let X be a quasi-separated algebraic stack. If k is an alge-braically closed field and x : Spec k → X is a point with non-affine stabilizer,then Aut( x ) → Aut ⊗ ( x ∗ ) is not injective. In particular, ω X (Spec k ) is notfaithful and X is not tensorial.Proof. By assumption, the stabilizer group scheme G x of x is not affine.Let H = ( G x ) ant be the largest anti-affine subgroup of G x ; then H is anon-trivial anti-affine group scheme over k and the quotient group scheme G x /H is affine [DG70, § III.3.8]. The induced morphism B k H → B k G x → X is thus quasi-affine by [Ryd11a, Thm. B.2].By [Bri09, Lem. 1.1], the morphism p : Spec k → B k H induces an equiv-alence of abelian tensor categories p ∗ : QCoh ( B k H ) → QCoh (Spec k ). SinceAut( p ) = H ( k ) = { id p } = Aut ⊗ ( p ∗ ), the functor ω B k H (Spec k ) is not faith-ful. Hence ω X (Spec k ) is not faithful by Lemma 4.3. (cid:3) We also have the following theorem.
Theorem 10.2.
Let X be a quasi-compact and quasi-separated algebraicstack with affine stabilizers. If k is a field and x : Spec k → X is anon-affine morphism, then there exists a field extension K/k and a point y : Spec K → X such that Isom( y, x ) → Hom ⊗ ( y ∗ , x ∗ ) is not surjective,where x denotes the K -point corresponding to x . In particular, ω X (Spec K ) is not full.Proof. To simplify notation, we let x = x . Since X has quasi-compactdiagonal, x is quasi-affine [Ryd11a, Thm. B.2]. By Lemma 4.3, we mayreplace X by S pec X ( x ∗ k ) and consequently assume that x is a quasi-compact ANNAKA DUALITY 31 open immersion and O X → x ∗ k is an isomorphism. In particular, x is asection to a morphism f : X → Spec k . Since x is not affine, it follows thatthere exists a closed point y disjoint from the image of x . In particular,there is a field extension K/k and a k -morphism y : Spec K → X whoseimage is a closed point disjoint from x .We now base change the entire situation by Spec K → Spec k . Thisresults in two morphisms x K , y K : Spec K → X ⊗ k K , where x K is a quasi-compact open immersion such that O X ⊗ k K ∼ = ( x K ) ∗ K and y K has image aclosed point disjoint from the image of x K . We replace X , k , x , and y by X ⊗ k K , K , x K , and y K respectively.Let G y ⊆ X be the residual gerbe associated to y , which is a closedimmersion. We define a natural transformation γ ∨ : x ∗ ⇒ y ∗ at k to be thecomposition x ∗ k ∼ = O X ։ O G y → y ∗ k and extend to all of QCoh (Spec k ) bytaking colimits. By adjunction, there is an induced natural transformation γ : y ∗ → x ∗ . A simple calculation shows that γ is a natural transformationof cocontinuous tensor functors. Since its adjoint γ ∨ is not an isomorphism, γ is not an isomorphism; thus γ is not realizable. The result follows. (cid:3) The following lemma is a variant of [Bha14, Ex. 4.12], which B. Bhattcommunicated to the authors.
Lemma 10.3.
Let k be an algebraically closed field and let G/k be an anti-affine group scheme of finite type. Let
Z/k be a regular scheme with a closedsubscheme C that is a nodal curve over k . Then there is a compatible systemof G -torsors E n → C [ n ] such that there does not exist a G -torsor E → Z that restricts to the E n s.Proof. Recall that G is smooth, connected and commutative [DG70, § III.3.8].Furthermore, by Chevalley’s theorem, there is an extension 0 → H → G → A →
0, where A is an abelian variety (of positive dimension) and H is affine.Let x A ∈ A ( k ) be an element of infinite order and let x ∈ G ( k ) be any liftof x A .Let e C be the normalization of C . Let F → C be the G -torsor obtained bygluing the trivial G -torsor on e C along the node by translation by x . Notethat the induced A -torsor F /H → C is not torsion as it is obtained bygluing along the non-torsion element x A .We may now lift F → C to G -torsors F n → C [ n ] . Indeed, the obstructionto lifting F n − to F n lies in Ext O C ( L g ∗ L • BG/k , I n /I n +1 ), where g : C → BG is the morphism corresponding to F → C and I is the ideal defining C in Z .Since G is smooth, the cotangent complex L • BG/k is concentrated in degree1 and since C is a curve, it has cohomological dimension 1. It follows thatthe obstruction group is zero.Now given a G -torsor F → Z , there is an induced A -torsor F/H → Z .Since Z is regular, the torsor F/H → Z is torsion in H ( Z, A ) [Ray70, XIII2.4 & 2.6]. Thus,
F/H → Z cannot restrict to F /H → C and the resultfollows. (cid:3) We now have the following theorem, which is a counterexample to [Aok06a,Thm. 1.1] and [Aok06b, Case I].
Theorem 10.4.
Let X → S be a quasi-separated morphism of algebraicstacks. If k is an algebraically closed field and x : Spec k → X is a point withnon-affine stabilizer, then there exists a morphism A k → S and a proper andflat family of curves Z → A k , where Z is regular, such that Hom A k ( Z, X × S A k ) is not algebraic.Proof. Let Q be the stabilizer group scheme of x and let G be the largestanti-affine subgroup scheme of Q ; thus, G is a non-trivial anti-affine groupscheme over k and the quotient group scheme Q/G is affine [DG70, § III.3.8].Let Z be a proper family of curves over T = A k = Spec k [ t ] with regulartotal space and a nodal curve C as the fiber over the origin; for example,take Z = Proj T ( k [ t ][ x, y, z ] / ( y z − x z − x − tz )) over T . Let T n = V ( t n +1 ),ˆ T = Spec ˆ O T, , Z n = Z × T T n , and ˆ Z = Z × T ˆ T . We now apply Lemma 10.3to C in ˆ Z and G . Since Z n = C [ n ] , this produces an element inlim ←− n Hom T ( Z, BG T )( T n ) = lim ←− n Hom( Z n , BG )that does not lift toHom T ( Z, BG T )( ˆ T ) = Hom( ˆ Z, BG ) . This shows that Hom T ( Z, BG T ) is not algebraic.By [Ryd11a, Thm. B.2], the morphism x factors as Spec k → BQ → Q → X , where Q is the residual gerbe, Q → X is quasi-affine and BQ → Q isaffine. Since Q/G is affine, it follows that the induced morphism BG → BQ → X is quasi-affine. By [HR15b, Thm. 2.3(ii)], the induced mor-phism Hom T ( Z, BG T ) → Hom T ( Z, X × S T ) is quasi-affine. In particular, ifHom T ( Z, X × S T ) is algebraic, then Hom T ( Z, BG T ) is algebraic, which is acontradiction. The result follows. (cid:3) The following theorem extends [Bha14, Ex. 4.12].
Theorem 10.5.
Let X be an algebraic stack with quasi-compact diagonal.If X does not have affine stabilizers, then there exists a noetherian two-dimensional regular ring A , complete with respect to an ideal I , such that X ( A ) → lim ←− X ( A/I n ) is not an equivalence of categories.Proof. Let x ∈ | X | be a point with non-affine stabilizer group. Arguing asin the proof of Theorem 10.4, there exists an algebraically closed field k ,an anti-affine group scheme G/k of finite type and a quasi-affine morphism BG → X . An easy calculation shows that it is enough to prove the theoremfor X = BG .Let A = k [ x, y ] and let A be the completion of A along the ideal I =( y − x − x ). Then Z = Spec A and C = Spec A/I satisfies the conditionsof Lemma 10.3 and we obtain an element in lim ←− n X ( A/I n ) that does not liftto X ( A ). (cid:3) Appendix A. Monomorphisms and stratifications
In this appendix, we introduce some notions and results needed for thefaithfulness part of Theorem 8.4 when T is not noetherian. This is essentialfor the proof of Corollary 1.4. ANNAKA DUALITY 33
Definition A.1.
We say that a finitely presented filtration ( X k ) of X (Defi-nition 8.1) is separating if the family { j nk : Y [ n ] k → X } k,n is separating [EGA,IV.11.9.1], that is, if the intersection T k,n ker (cid:0) O X → ( j nk ) ∗ O Y [ n ] k (cid:1) is zero asa lisse-´etale sheaf. Lemma A.2.
Every finitely presented filtration ( X k ) on X is separating ifeither (i) X is noetherian; or (ii) X has no embedded (weakly) associated point.If X is noetherian with a filtration ( X k ) and X ′ → X is flat, then ( X k × X X ′ ) is a separating filtration on X ′ .Proof. As the question is smooth-local, we can assume that X and X ′ areaffine schemes. If X is noetherian, then by primary decomposition thereexists a separating family ` mi =1 Spec A i → X where the A i are artinian. Asevery Spec A i factors through some Y [ n ] k , it follows that ( X k ) is separating.In general, { Spec O X,x → X } x ∈ Ass( X ) is separating [Laz64, 1.2, 1.5, 1.6]. If x is a non-embedded associated point, then Spec O X,x is a one-point schemeand factors through some Y [ n ] k and the first claim follows.For the last claim, we note that a finite number of the infinitesimal neigh-borhoods of the strata suffices in the noetherian case and that flat morphismspreserve kernels and finite intersections. (cid:3) Proposition A.3.
Let X be an algebraic stack with a finitely presentedfiltration ( X k ) . Let f : Z → X be a morphism locally of finite type. If f | Y [ n ] k is an isomorphism for every k and n , then f is a surjective closedimmersion. If in addition ( X k ) is separating, then f is an isomorphism.Proof. Note that f is a surjective and quasi-compact monomorphism. Wewill prove that f is a closed immersion by induction on the number ofstrata r . If r = 0, then X = ∅ and there is nothing to prove. If r = 1,then X = X [ n ]1 = Y [ n ]1 for sufficiently large n and the result follows. If r ≥ U = X \ X . By the induction hypothesis, f | U is a surjective closedimmersion.It is enough to show that f is a closed immersion in a neighborhood ofevery x ∈ | X | . This can be checked locally in the ´etale topology and wemay thus assume that Z = Z ∐ Z where Z → X is a closed immersionand Z ∩ f − ( x ) = ∅ . Note that f ( Z ) ∩ U and f ( Z ) ∩ U are disjoint closedand open subsets.It is further enough to show that f is a closed immersion after replacing X with either X , f ( Z ) ∩ U or f ( Z ) ∩ U . In the first and second case, f is certainly a closed immersion. In the third case, f ( Z ) is set-theoreticallycontained in X . Let W = f ( Z ) ∩ X ; this is an open and closed substackof X . Thus, f | Z : Z → X factors through W [ n ] for sufficiently large n .By hypothesis, this means that Z ∼ = W [ n ] ∼ = W [ N ] for all sufficiently large n and all N ≥ n . This implies that W [ n ] ֒ → X is an open immersion andwe have proved that f is a closed immersion in a neighborhood of x . Theresult follows.The last claim is obvious. (cid:3) The following example illustrates that a closed immersion f : Z → X as in Proposition A.3 need not be an isomorphism even if f is of finitepresentation. Example A.4.
Let A = k [ x, z , z , . . . ] / ( xz , { z k − xz k +1 } k ≥ , { z i z j } i,j ≥ )and B = A/ ( z ). Then A/ ( x n ) = k [ x ] / ( x n ) = B/ ( x n ) and A x = k [ x ] x = B x but the surjection A → B is not an isomorphism. Appendix B. A relative boundedness result for Hom stacks
Here we prove the following relative boundedness result for Hom stacks.
Theorem B.1.
Let f : Z → S be a proper, flat and finitely presented mor-phism of algebraic stacks. Let X and Y be algebraic stacks that are locallyof finite presentation and quasi-separated over S and have affine stabilizersover S . Let g : X → Y be a finitely presented S -morphism. If g has affinefibers, then Hom S ( Z, g ) : Hom S ( Z, X ) → Hom S ( Z, Y ) is of finite presentation. If in addition g : X → Y is a group, then Hom S ( Z, g ) is a group with affine fibers. Theorem B.1 is used in Theorems 1.2 and 1.3 to establish the quasi-compactness of the diagonal of Hom-stacks and Weil restrictions.Without using Theorem B.1, the proof of Theorem 1.2 give the algebraic-ity of the Hom-stacks and that they have quasi-separated diagonals. Inthe setting of Theorem B.1, we may conclude that Hom S ( Z, g ) is a quasi-separated morphism of algebraic stacks that are locally of finite presentationover S . It remains to prove that the morphism Hom S ( Z, g ) is quasi-compact.
Preliminary reductions. If W and T are algebraic stacks over S , let W T = W × S T ; similarly for morphisms between stacks over S . We will usethis notation throughout this appendix.As the question is local on S , we may assume that S is an affine scheme.We may also assume that X and Y are of finite presentation over S since it isenough to prove the theorem after replacing Y with an open quasi-compactsubstack and X with its inverse. By standard approximation results, wemay then assume that S is of finite type over Spec Z . For the remainder ofthis article, all stacks will be of finite presentation over S and hence excellentwith finite normalization.By noetherian induction on S , to prove that Hom S ( Z, g ) is quasi-compact,we may assume that S is integral and replace S with a suitable dense opensubscheme. Moreover, we may also replace Z → S with the pull-back alonga dominant map S ′ → S . Recall that there exists a field extension K ′ /K ( S )such that ( Z K ′ ) red (resp. ( Z K ′ ) norm ) is geometrically reduced (resp. geomet-rically normal) over K ′ . After replacing S with a dense open subset of thenormalization in K ′ , we may thus assume that(i) Z red → S is flat with geometrically reduced fibers; and(ii) Z norm → S is flat with geometrically normal fibers;since these properties are constructible [EGA, IV.9.7.7 (iii) and 9.9.4 (iii)].We now prove three reduction results. Throughout, we will assume thefollowing: ANNAKA DUALITY 35 • Z is proper and flat over S , • X and Y are finitely presented algebraic stacks over S with affinestabilizers, and • g : X → Y is a representable morphism over S .Our first reduction result is similar to [Ols06b, Lem. 5.11]. Lemma B.2. If Hom S ′ ( Z ′ , g S ′ ) is quasi-compact for every scheme S ′ , mor-phism S ′ → S and nil-immersion Z ′ → Z S ′ such that Z ′ → S ′ is proper andflat with geometrically reduced fibers, then Hom S ( Z, g ) is quasi-compact.Proof. Assume that the condition holds. To prove that Hom S ( Z, g ) is quasi-compact, we may assume that S is integral. We may also assume that Z red → S has geometrically reduced fibers. Pick a sequence of square-zeronil-immersions Z red = Z ֒ → Z ֒ → . . . ֒ → Z n = Z . After replacing S witha dense open subset, we may assume that all the Z i → S are flat. Thus, itsuffices to show that if j : Z → Z is a square-zero closed immersion where Z is flat over S and Hom S ( Z , g ) is quasi-compact, then Hom S ( Z, g ) is quasi-compact. Now argue as in [Ols06b, Lem. 5.11], but this time using thedeformation theory of [Ols06a, Thm. 1.5] and the Semicontinuity Theoremof [Hal14a, Thm. A]. (cid:3)
Before we proceed, we make the following observation: fix an S -scheme T and an S -morphism y : Z T → Y . This corresponds to a map T → Hom S ( Z, Y ). The pullback of Hom S ( Z, g ) along this map is isomorphic tothe Weil restriction R Z T /T ( X × g,Y,y Z T ), which we will denote as H Z/S,g ( y ).Note that our hypotheses guarantee that H Z/S,g ( y ) is locally of finite typeand quasi-separated over T .The second reduction is for a (partial) normalization. Lemma B.3. If Hom S ′ ( Z ′ , g S ′ ) is quasi-compact for every scheme S ′ , mor-phism S ′ → S and finite morphism Z ′ → Z S ′ such that Z ′ → S ′ is properand flat with geometrically normal fibers, then Hom S ( Z, g ) is quasi-compact.Proof. By Lemma B.2, we may assume that Z → S is flat with geometricallyreduced fibers. We will use induction on the maximal fiber dimension d of Z → S . After modifying S , we may assume that W := Z norm → S is flatwith geometrically normal fibers. Let Z ֒ → Z and W ֒ → W be the closedsubstacks given by the conductor ideal of W → Z .After replacing S with a dense open subset, we may assume that Z → S and W → S are flat and that W → Z is an isomorphism over an opensubset U ⊆ Z that is dense in every fiber. In particular, since Z ∩ U = ∅ ,the dimensions of the fibers of Z → S are strictly smaller than d . Thus, byinduction we may assume that Hom S ( Z , g ) is quasi-compact. But W (cid:31) (cid:127) i / / h (cid:15) (cid:15) W h (cid:15) (cid:15) Z (cid:31) (cid:127) j / / Z (cid:3) is a bicartesian square and remains so after arbitrary base change over S since W → S is flat. Indeed, that it is cartesian is [Hal14b, Lem. A.3(i)].That it is cocartesian and the commutes with arbitrary base change over S follows from the arguments of [Hal14b, Lem. A.4, A.8] and the existence ofpinchings of algebraic spaces [Kol11, Thm. 38].It remains to prove that H Z/S,g ( y ) → T is quasi-compact, where T is anintegral scheme of finite type over S and y : Z T → Y is a morphism. Thebicartesian square above implies that H Z/S,g ( y ) ≃ H Z /S,g ( yj ) × H W /S,g ( yhi ) H W/S,g ( yh ) . The result follows, since H Z /S,g ( yj ) and H W/S,g ( yh ) are quasi-compact and H W /S,g ( yhi ) is quasi-separated. (cid:3) We have the following variant of h -descent [Ryd10, Thm. 7.4]. Lemma B.4.
Let S be an algebraic stack, let T be an algebraic S -stackand let g : T ′ → T be a universally subtrusive (e.g., proper and surjective)morphism of finite presentation such that g is flat over an open substack U ⊆ T . If T is weakly normal in U (e.g., T normal and U open dense),then for every representable morphism X → S , the following sequence ofsets is exact: X ( T ) / / X ( T ′ ) / / / / X ( T ′ × T T ′ ) where X ( T ) = Hom S ( T, X ) etc.Proof. It is enough to prove that given a morphism f : T ′ → X such that f ◦ π = f ◦ π : T ′ × T T ′ → X , there exists a unique morphism h : T → X suchthat f = h ◦ g . By fppf-descent over U , there is a unique h | U : U → X suchthat f | g − ( U ) = h | U ◦ g | g − ( U ) . Consider the morphism ˜ g : e T ′ = T ′ ∐ U → T .The morphism ˜ f = ( f, h | U ) : e T ′ → X satisfies ˜ f ◦ ˜ π = ˜ f ◦ ˜ π where ˜ π i denotes the projections of e T ′ × X e T ′ → e T ′ . By assumption, ˜ g is universallysubtrusive and weakly normal. Thus, by h -descent [Ryd10, Thm. 7.4], wehave an exact sequence X ( T ) / / X ( e T ′ ) / / / / X (cid:0) ( e T ′ × T e T ′ ) red (cid:1) . Indeed, by smooth descent we can assume that S , T and e T ′ are schemes sothat [Ryd10, Thm. 7.4] applies. We conclude that ˜ f comes from a uniquemorphism h : T → X . (cid:3) We now have our last general reduction result.
Proposition B.5.
Let w : W → Z be a proper surjective morphism over S .Assume that Z → S has geometrically normal fibers and W → S is flat. If Hom S ( W, g ) is quasi-compact, then so is Hom S ( Z, g ) .Proof. We may assume that S is an integral scheme. After replacing S withan open subscheme, we may also assume that W → Z is flat over an opensubset U ⊆ Z that is dense in every fiber over S and W × Z W is flat over S . It remains to prove that H Z/S,g ( y ) → T is quasi-compact, where T isan integral scheme of finite type over S and y : Z T → Y is a morphism. Byassumption, H W/S,g ( yw ) → T is quasi-compact. Now consider the sequence: H Z/S,g ( y ) / / H W/S,g ( yw ) / / / / H W × Z W/S,g ( yv ) , ANNAKA DUALITY 37 where v : W × Z W → Z is the natural map. There is a canonical mor-phism ϕ : H Z/S,g ( y ) → E , where E denotes the equalizer of the parallelarrows. Since H W/S,g ( yw ) is quasi-compact (and H W × V W/S,g ( yv ) is quasi-separated), the equalizer E is quasi-compact. It is thus enough to show that ϕ is quasi-compact. Thus, pick a scheme T ′ and a morphism T ′ → E andlet us show that H Z/S,g ( y ) × E T ′ is quasi-compact.By noetherian induction on T ′ , we may assume that T ′ is normal. Themorphism T ′ → E gives an element of Hom Y ( W T ′ , X ) such that the two im-ages in Hom Y ( W T ′ × Z T ′ W T ′ , X ) coincide. Noting that Z T ′ is normal, LemmaB.4 applies to W T ′ → Z T ′ and gives a unique element in Hom Y ( Z T ′ , X ) =Hom T ( T ′ , H Z/S,g ( y )). Thus, the morphism ϕ T ′ : H Z/S,g ( y ) × E T ′ → T ′ hasa section. Repeating the argument with T ′ = Spec κ ( t ′ ) for every point t ′ ∈ T ′ , we see that ϕ T ′ is injective, so the section is surjective. It followsthat H Z/S,g ( y ) × E T ′ is quasi-compact. (cid:3) Proof of the main result.
Proof of Theorem B.1.
As usual, we may assume that S is an affine integralscheme. By Lemma B.3, we may in addition assume that Z → S hasgeometrically normal fibers. Let W → Z be a proper surjective morphismwith W a projective S -scheme [Ols05]. By replacing S with a dense open,we may assume that W → S is flat. By Proposition B.5, we may replace Z with W and assume that Z is a (projective) scheme. Repeating the firstreduction, we may still assume that Z → S has geometrically normal fibers.As before, it remains to prove that H Z/S,g ( y ) → T is quasi-compact, where T is an integral S -scheme of finite type and y : Z T → Y is an S -morphism.Hence, it suffices to prove the following claim. Claim:
Let S be integral. If Z → S is projective with geometrically normalfibers and q : Q → Z is representable with affine fibers, then R Z/S ( Q ) → S is quasi-compact. Proof of Claim:
Let Q = Spec Z ( q ∗ O Q ) and let Q → Q → Z be the inducedfactorization. Since Q → Z has affine fibers, Q → Q is an isomorphismover an open dense subset U ⊆ Z . After replacing S with a dense opensubscheme, we may assume that U is dense in every fiber over S . Since R Z/S ( Q ) → S is affine [HR15b, Thm. 2.3(i)], it is enough to prove that R Z/S ( Q ) → R Z/S ( Q ) is quasi-compact. We may thus replace Q , Z , U and S with Q × Q ( Z × S R Z/S ( Q )), Z × S R Z/S ( Q ), U × S R Z/S ( Q ) and R Z/S ( Q ).We may thus assume that Q → Z is an isomorphism over U .Since Q is an algebraic space, there exists a finite surjective morphism˜ Q → Q such that ˜ Q is a scheme. In particular, there is a finite field extension L/K ( U ) such that the normalization of Q in L is a scheme. Take a splittingfield L ′ /L and let Z ′ be the normalization of Z in L ′ . Then Q ′ := ( Q × Z Z ′ ) norm = Q norm /L ′ is a scheme. By replacing S with a normalization in anextension of K ( S ) and shrinking, we may assume that Z ′ → S and Q ′ → S are flat with geometrically normal fibers. By Proposition B.5, it is enoughto prove that R Z ′ /S ( Q × Z Z ′ ) is quasi-compact.There is a natural morphism R Z ′ /S ( Q ′ ) → R Z ′ /S ( Q × Z Z ′ ), which weclaim is surjective. To see this, we may assume that S is the spectrum ofan algebraically closed field. Then Z ′ and Q ′ are normal and any section Z ′ → Q × Z Z ′ lifts uniquely to a section Z ′ → Q ′ . Indeed, Z ′ × Q × Z Z ′ Q ′ → Z ′ is finite and an isomorphism over U , hence has a canonical section. We canthus replace Q and Z with Q ′ and Z ′ and assume that Q is a scheme.Since Q is a scheme, it is locally separated; hence, there is a U -admissibleblow-up Z ′ → Z such that the strict transform Q ′ → Z ′ of Q → Z is´etale [RG71, Thm. 5.7.11]. After shrinking S , we may assume that Z ′ → S is flat. Then since U ⊆ Z ′ remains dense after arbitrary pull-back over S ,we have that R Z ′ /S ( Q × Z Z ′ ) = R Z ′ /S ( Q ′ ). Replacing Q → Z with Q ′ → Z ′ (Proposition B.5), we may thus assume that Q → Z in addition is ´etale.Finally, we note that the ´etale morphism Q → Z corresponds to a con-structible sheaf on Z ´Et and that R Z/S ( Q ) is nothing but the ´etale sheaf f ´Et , ∗ Q . By a special case of the proper base change theorem [SGA4 ,XIV.1.1], f ´Et , ∗ Q is constructible, so R Z/S ( X ) → S is of finite presentation.For the second part of the theorem on groups: let T be the the spectrumof an algebraically closed field and let y : Z T → Y be a morphism. By thefirst part H Z/S,g ( y ) ≃ R Z T /T ( Q ) is then a group scheme G of finite typeover T , where Q = X × Y Z T . Let K = G ant be the largest anti-affinesubgroup of G ; it is normal, connected and smooth and the quotient G/K is affine [DG70, § III.3.8].The universal family G × T Z T → Q is a group homomorphism and inducesa group homomorphism K × T Z T → Q . It is enough to show that this factorsthrough the unit section of Q → Z T , because this forces K = 0.Note that for every stack W → T , the pull-back H × T W → W is ananti-affine group in the sense that the push-forward of O H × T W is O W (flatbase change). Since Q → Z T has affine fibers, there is a finitely presentedfiltration ( Z i ) of Z T with strata V [ n ] i over which Q × Z T V [ n ] i → V [ n ] i is affine.Since K × T V [ n ] i is anti-affine, it follows that K × T V [ n ] i → Q × Z T V [ n ] i factorsthrough the unit section V [ n ] i → Q × Z T V [ n ] i .Let E be the equalizer of K × T Z T → Q and the constant map K × T Z T → Q to the unit. The above discussion shows that the monomorphism E → K × T Z T is an isomorphism over every strata V [ n ] i , hence an isomor-phism (Proposition A.3, using that the filtration is separating since Z T isnoetherian). (cid:3) References [AB05] V. Alexeev and M. Brion,
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Mathematical Sciences Institute, The Australian National University, Ac-ton ACT 2601, Australia
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