Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited
aa r X i v : . [ m a t h . AG ] F e b Quotient stacks and equivariant étale cohomology algebras:Quillen’s theory revisited
Luc Illusie Weizhe Zheng
To the memory of Daniel Quillen
Abstract
Let k be an algebraically closed field. Let Λ be a noetherian commutative ring annihilatedby an integer invertible in k and let ℓ be a prime number different from the characteristicof k . We prove that if X is a separated algebraic space of finite type over k endowed with anaction of a k -algebraic group G , the equivariant étale cohomology algebra H ∗ ([ X/G ] , Λ), where[
X/G ] is the quotient stack of X by G , is finitely generated over Λ. Moreover, for coefficients K ∈ D + c ([ X/G ] , F ℓ ) endowed with a commutative multiplicative structure, we establish astructure theorem for H ∗ ([ X/G ] , K ), involving fixed points of elementary abelian ℓ -subgroupsof G , which is similar to Quillen’s theorem [36, Theorem 6.2] in the case K = F ℓ . One keyingredient in our proof of the structure theorem is an analysis of specialization of points ofthe quotient stack. We also discuss variants and generalizations for certain Artin stacks. Introduction
In [36], Quillen developed a theory for mod ℓ equivariant cohomology algebras H ∗ G ( X, F ℓ ), where ℓ is a prime number, G is a compact Lie group, and X is a topological space endowed with an actionof G . Recall that, for r ∈ N , an elementary abelian ℓ -group of rank r is defined to be a groupisomorphic to the direct product of r cyclic groups of order ℓ [36, Section 4]. Quillen showed that H ∗ G ( X, Λ) is a finitely generated Λ-algebra for any noetherian commutative ring Λ [36, Corollary2.2] and established structure theorems ([36, Theorem 6.2], [37, Theorem 8.10]) relating the ringstructure of H ∗ G ( X, F ℓ ) to the elementary abelian ℓ -subgroups A of G and the components of thefixed points set X A . We refer the reader to [24, Section 1] for a summary of Quillen’s theory.In this article, we establish an algebraic analogue. Let k be an algebraically closed field ofcharacteristic = ℓ and let Λ be noetherian commutative ring annihilated by an integer invertible in k . Let G be an algebraic group over k ( not necessarily affine) and let X be a separated algebraicspace of finite type over k endowed with an action of G . We consider the étale cohomology ring H ∗ ([ X/G ] , Λ) of the quotient stack [
X/G ]. One of our main results is that this ring is a finitelygenerated Λ-algebra (Theorem 4.6) and the ring homomorphism H ∗ ([ X/G ] , F ℓ ) → lim ←− A H ∗ ( BA, F ℓ )given by restriction maps is a uniform F -isomorphism (Theorem 6.11), i.e. has kernel and cokernelkilled by a power of F : a a ℓ (see Definition 6.10 for a review of this notion introduced byQuillen [36, Section 3]). Here A is the category of pairs ( A, C ), where A is an elementary abelian ℓ -subgroup of G and C is a connected component of X A . The morphisms ( A, C ) → ( A ′ , C ′ ) of A are given by elements g ∈ G such that Cg ⊃ C ′ and g − Ag ⊂ A ′ . We also establish a generalization(Theorem 6.17) for H ∗ ([ X/G ] , K ), where K ∈ D + c ([ X/G ] , F ℓ ) is a constructible complex of sheaveson [ X/G ] endowed with a commutative ring structure.A key ingredient in Quillen’s original proofs is the continuity property [36, Proposition 5.6]. Inthe algebraic setting, this property is replaced by an analysis of the specialization of points of thequotient stack [
X/G ]. In order to make sense of this, we introduce the notions of geometric pointsand of ℓ -elementary points of Artin stacks. Our structure theorems for equivariant cohomologyalgebras are consequences of the following general structure theorem (Theorem 8.3): if X = [ X/G ]1r X is a Deligne-Mumford stack of finite presentation and finite inertia over k , and if K ∈ D + c ( X , F ℓ ) is endowed with a commutative ring structure, then the ring homomorphism H ∗ ( X , K ) → lim ←− x : S→X H ∗ ( S , x ∗ K )given by restriction maps is a uniform F -isomorphism. Here the limit is taken over the categoryof ℓ -elementary points of X .In [26] we established an algebraic analogue [26, Theorem 8.1] of a localization theorem ofQuillen [36, Theorem 4.2], which he had deduced from his structure theorems for equivariantcohomology algebras. This was one of the motivations for us to investigate algebraic analogues ofthese theorems. We refer the reader to [25] for a report on the present article and on some resultsof [26].In Part I we review background material on quotient and classifying stacks (Section 1), andcollect results on the cohomology of Artin stacks (Section 2) that are used at different places in thisarticle. The ring structures of the cohomology algebras we are considering reflect ring structureson objects of derived categories. We discuss this in Section 3.The reader familiar with the general nonsense recalled in Part I could skip it and move directlyto Part II, which contains the main results of the paper. In Section 4, we prove the above-mentionedfiniteness theorem (Theorem 4.6) for equivariant cohomology algebras. One key step of the proofamounts to replacing an abelian variety by its ℓ -divisible group, which was communicated to usby Deligne. In Section 5, we present a crucial result on the finiteness of orbit types, which is ananalogue of [36, Lemma 6.3] and was communicated to us by Serre.In Section 6, we state the above-mentioned structure theorems (Theorems 6.11, 6.17) for equiv-ariant cohomology algebras. In Section 7, we introduce and discuss the notions of geometric pointsand of ℓ -elementary points of Artin stacks. Using them we state in Section 8 the main result ofthis paper, the structure theorem (Theorem 8.3) for cohomology algebras of certain Artin stacks,and show that it implies the structure theorems of the equivariant case. In Section 9, we establishsome Künneth formulas needed in the proof of Theorem 8.3, which is given in Section 10. Finally,in Section 11 we prove an analogue of Quillen’s stratification theorem [37, Theorems 10.2, 12.1]for the reduced spectrum of mod ℓ étale equivariant cohomology algebras.The results of this paper have applications to the structure of varieties of supports. We hopeto return to this in a future article. Acknowledgments
We thank Pierre Deligne for the proof of the finiteness theorem (Theorem 4.6) in the general caseand Jean-Pierre Serre for communicating to us the results of Section 5. We are grateful to MichelBrion for discussions on the cohomology of classifying spaces and Michel Raynaud for discussionson separation issues. The second author thanks Ching-Li Chai, Johan de Jong, Yifeng Liu, MartinOlsson, David Rydh, and Yichao Tian for useful conversations. We thank the referees for theircareful reading of the manuscript and many helpful comments.Part of this paper was written during a visit of both authors to the Korea Institute for AdvancedStudy in Seoul in January 2013 and a visit of the first author to the Morningside Center ofMathematics, Chinese Academy of Sciences in Beijing in February and March 2013. Warm thanksare addressed to these institutes for their hospitality and support.The second author was partially supported by China’s Recruitment Program of Global Experts;National Natural Science Foundation of China Grant 11321101; Hua Loo-Keng Key Laboratory ofMathematics, Chinese Academy of Sciences; National Center for Mathematics and InterdisciplinarySciences, Chinese Academy of Sciences.
Conventions
We fix a universe U , which we will occasionally enlarge. We say “small” instead of “ U -small” whenthere is no ambiguity. We say that a category is essentially small (resp. essentially finite ) if it isequivalent to a small (resp. finite) category. Schemes are assumed to be small. Presheaves takevalues in the category of U -sets. For any category C , we denote by b C the category of presheaves2n C , which is a U -topos if C is essentially small. If f : C → D is a fibered category, we denote by C ( U ) (or sometimes C U ) the fiber category of f over an object U of D .By a stack over a U -site C we mean a stack in groupoids over C [44, 02ZI] whose fiber categoriesare essentially small. By a stack, we mean a stack over the big fppf site of Spec( Z ). Unlike [31], wedo not assume algebraic spaces and Artin stacks to be quasi-separated. We say that a morphism f : X → Y of stacks is representable (this property is called “representable by an algebraic space”in [44, 02ZW]) if for every scheme U and every morphism y : U → Y , the 2-fiber product U × y, Y ,f X is representable by an algebraic space. By an Artin stack (resp.
Deligne-Mumford stack ), we meanan “algebraic stack” (resp. Deligne-Mumford stack) over Spec( Z ) in the sense of [44, 026O] (resp.[44, 03YO]), namely a stack X such that the diagonal ∆ X : X → X × X is representable and suchthat there exists an algebraic space X and a smooth (resp. étale) surjective morphism X → X .By an algebraic group over a field k , we mean a group scheme over k of finite type. Unlessotherwise stated, groups act on the right. Contents
I Preliminaries 3
II Main results 21
Part I
Preliminaries
Classically, if G is a compact Lie group, a classifying space BG for G is the base of a contractible(right) G -torsor P G . Such a classifying space exists and is essentially unique (up to homotopyequivalence). If X is a G -space (i.e. a topological space endowed with a continuous (right) actionof G ), one can twist X by P G and get a space
P G ∧ G X , defined as the quotient of P G × X by The fiber categories of prestacks over C are also assumed to be essentially small. G , (( p, x ) , g ) ( pg, xg ). This space P G ∧ G X is a fiber bundle over BG offiber X , and P G × X is a G -torsor over P G ∧ G X . If Λ is a ring, the equivariant cohomology of X with value in Λ is defined by H ∗ G ( X, Λ) := H ∗ ( P G ∧ G X, Λ) ≃ H ∗ ( BG, Rπ ∗ Λ)where π : P G ∧ G X → BG is the projection. The functorial properties of this cohomology, intro-duced by Borel, are discussed by Quillen in [36, Section 1].A well-known similar formalism exists in algebraic geometry, with classifying spaces replacedby classifying stacks. We review this formalism in this section. Construction 1.1.
Let C be a category in which finite limits are representable. We define thecategory Eq( C )of equivariant objects in C as follows. The objects of Eq( C ) are pairs ( X, G ) consisting of a groupobject G of C and an object X of C endowed with an action of G , namely a morphism X × G → X satisfying the usual axioms for composition and identity. A morphism ( X, G ) → ( Y, H ) in Eq( C ) isa pair ( f, u ) consisting of a homomorphism u : G → H and a u -equivariant morphism f : X → Y .Here the u -equivariance of f is the commutativity of the following diagram in C : X × G / / f × u (cid:15) (cid:15) X f (cid:15) (cid:15) Y × H / / Y. While Eq( C ) is a category, groupoids in C form a (2,1)-category Grpd( C ) . We regard groupoids X • in C as internal categories, consisting of two objects X and X of C ,called respectively the object of objects and the object of morphisms, together with four morphismsin C , e : X → X , s, t : X → X , m : X × s X ,X ,t X X → X , called respectively identity, source, target, and composition. A 1-morphism of groupoids f • : X • → Y • is an internal functor between the underlying internal categories, namely a pair of morphisms f : X → Y , f : X → Y , compatible with e , s , t , m . For 1-morphisms of groupoids f • , g • : X • → Y • , a 2-morphism f • → g • is an internal natural isomorphism, namely, a morphism r : X → Y of C such that s Y r = f , t Y r = g , and m Y ( g , rs X ) = m Y ( rt X , f ). The last identity can be statedinformally as for any ( u : a → b ) ∈ X , g ( u ) r ( a ) = r ( b ) f ( u ).We define a functor(1.1.1) Eq( C ) → Grpd( C ) . as follows. To an object ( X, G ) of Eq( C ), we assign a groupoid in C ( X, G ) • with ( X, G ) = X , ( X, G ) = X × G , e ( x ) = ( x, s ( x, g ) = xg , t ( x, g ) = x , and compositiongiven by ( x, g )( xg, h ) = ( x, gh ). The inverse-assigning morphism is ( x, g ) ( xg, g − ). Here wefollow the conventions of [31, 3.4.3] (see also [44, 0444] where groups act on the left). A morphism( f, u ) : ( X, G ) → ( Y, H ) in Eq( C ) gives a morphism of groupoids ( f, u ) • : ( X, G ) • → ( Y, H ) • ,( f, u ) = f , ( f, u ) = f × u : ( x, g ) ( f ( x ) , u ( g )).The functor (1.1.1) is faithful, but not fully faithful. The maximal 2-subcategory Grpd Eq ( C )of Grpd( C ) spanned by the objects in the image of (1.1.1) can be described as follows. Proposition 1.2.
Let ( X, G ) , ( Y, H ) , and ( Z, I ) be objects of Eq( C ) . A (2,1)-category is a 2-category whose 2-morphisms are invertible. a) For any morphism of groupoids ϕ = ( ϕ , ϕ ) : ( X, G ) • → ( Y, H ) • , there exist a unique pairof morphisms f : X → Y , u : X × G → H such that ϕ ( x, g ) = ( f ( x ) , u ( x, g )) , and the pair ( f, u ) satisfies the following relations:(i) f is u -equivariant , i.e. f ( xg ) = f ( x ) u ( x, g ) ,(ii) u ( x, g ) u ( xg, g ′ ) = u ( x, gg ′ ) .Conversely, any pair ( f, u ) satisfying (i), (ii) defines a morphism of groupoids ϕ • . Moreover,if ( a, u ) : ( X, G ) • → ( Y, H ) • and ( b, v ) : ( Y, H ) • → ( Z, I ) • are morphisms of groupoids, thecomposition is given by ( ba, w ) , where w : X × G → I is given by w ( x, g ) = v ( a ( x ) , u ( x, g )) .(b) Let ϕ i = ( f i , u i ) : ( X, G ) • → ( Y, H ) • ( i = 1 , ) be 1-morphisms of groupoids. Then a 2-morphism from ϕ to ϕ is a morphism r : X → H satisfying the relations(i) f ( x ) = f ( x ) r ( x ) ,(ii) r ( x ) u ( x, g ) = u ( x, g ) r ( xg ) .Composition of 2-morphisms is given by multiplication in H . We will sometimes call a morphism u : X × G → H satisfying (a) (ii) a crossed homomorphism . Proof.
In (a), the uniqueness of ( f, u ) are clear, while the existence (resp. (i), resp. (ii)) expressesthe compatibility of ϕ with the target (resp. source, resp. composition) morphism. The otherstatements are straightforward. Definition 1.3.
We say that a pseudofunctor F : C → D between 2-categories is faithful (resp. fully faithful ) if for every pair of objects X and Y in C , the functor Hom C ( X, Y ) → Hom D ( F X, F Y )induced by F is fully faithful (resp. an equivalence of categories). We say that F is essentiallysurjective if for every object Y of D , there exists an object X of C and an equivalence F X ≃ Y in D . Construction 1.4.
Let E be a U -topos (we will be mostly interested in the case where E is thetopos of fppf sheaves on some algebraic space), endowed with its canonical topology. A groupoid X • in E defines a category [ X • ] ′ fibered in groupoids over E whose fiber at U is X • ( U ). This isan E -prestack, and, as in [31, 3.4.3], we denote the associated E -stack [44, 02ZP] by [ X • ]. If π denotes the canonical composite morphism π : X → [ X • ] ′ → [ X • ] , the groupoid can be recovered from π : there is a natural isomorphism(1.4.1) X ∼ −→ X × [ X • ] X identifying the projections p , p : X × [ X • ] X → X with s and t , and identifying the secondprojection id × π × id : X × [ X • ] X × [ X • ] X → X × [ X • ] X with m . Here X × [ X • ] X denotes thesheaf carrying U to the set of isomorphism classes of triples ( x, y, α ), x, y ∈ X ( U ), α : π ( x ) ≃ π ( y ).More generally, there is a natural isomorphism of simplicial objects(1.4.2) Ner( X • ) ∼ −→ cosk ( π )between the nerve of the groupoid X • and the 0-th coskeleton of π .We denote by Stack( E ) (resp. PreStack( E )) the (2,1)-category of E -stacks ( E -prestacks). Thepseudofunctor Grpd( E ) → PreStack( E ) sending X • to [ X • ] ′ is fully faithful and the pseudofunctorPreStack( E ) → Stack( E ) sending an E -prestack to its associated E -stack is faithful. Therefore,the composite pseudofunctor(1.4.3) Grpd( E ) → Stack( E )sending X • to its associated E -stack [ X • ] is faithful. In other words, if X • , Y • are groupoids in E ,and ϕ i : X • → Y • ( i = 1 ,
2) is a morphism of groupoids, then the natural mapHom( ϕ , ϕ ) → Hom([ ϕ ] , [ ϕ ])is bijective. However, in general, not every morphism f : [ X • ] → [ Y • ] is of the form [ ϕ ] for amorphism of groupoids ϕ : X • → Y • (see Remark 1.7 below). On the other hand, (1.4.3) isessentially surjective. 5 otation 1.5. In the case of the groupoid (
X, G ) • associated with a G -object X of E , the stack[( X, G ) • ] is denoted by(1.5.1) [ X/G ]and called the quotient stack of X by G . For X = e the final object of E (with the trivial actionof G ), it is called the classifying stack of G and denoted by(1.5.2) BG := [ e/G ] . Recall ([31, 2.4.2], [44, 04WM]) that the projection X → [ X/G ] makes X into a universal G -torsor over [ X/G ], i.e. for U in E , the groupoid [ X/G ]( U ) is canonically equivalent to the categoryof pairs ( P, a ), where P is a right G U -torsor and a is a G -equivariant morphism from P to X ;morphisms from ( P, a ) to (
Q, b ) are G -equivariant morphisms c : P → Q such that a = bc .The action of G on X is recovered from π : the isomorphism (1.4.1) takes the form(1.5.3) X × G ∼ −→ X × [ X/G ] X, identifying the projections p , p with ( x, g ) xg , ( x, g ) x .For X = e , BG ( U ) is the groupoid of G -torsors on U for U in E , which justifies the terminology“classifying stack”. For general X , the projection [ X/G ] → BG induces X → e by the base change B { } → BG , so that one can think of [ X/G ] → BG as a “fibration” with fiber X . In other words,[ X/G ] plays the role of the object
P G ∧ G X recalled at the beginning of Section 1.In order to describe morphisms from [ X/G ] to [
Y /H ] associated to morphisms of groupoidsfrom (
X, G ) • to ( Y, H ) • , we need to introduce the following notation. Let ( X, G ) be an object ofEq( E ), and let u : X × G → H be a crossed homomorphism (Proposition 1.2). We denote by(1.5.4) X ∧ G,u H the quotient of X × H by G acting by ( x, h ) g = ( xg, u ( x, g ) − h ). This is an H -object of E , theaction of H on it being deduced from its action by right translations on X × H . For any H -object Y of E , the map(1.5.5) Hom u ( X, Y ) → Hom H ( X ∧ G,u
H, Y )sending a u -equivariant morphism f (Proposition 1.2 (a) (i)) to the morphism f u : ( x, h ) f ( x ) h is bijective.When u : X × G → H is defined by u ( x, g ) = u ( g ) for a group homomorphism u : G → H , X ∧ G,u H coincides with the usual contracted product [20, Définition III.1.3.1], i.e. the quotient of X × H by the diagonal action of G , ( x, h ) g := ( xg, u ( g ) − h ).The following proposition, whose verification is straightforward, describes the restriction of(1.4.3) to Grpd Eq ( E ). Proposition 1.6.
Let ( X, G ) and ( Y, H ) be objects of Eq( E ) .(a) Let ( f, u ) : ( X, G ) • → ( Y, H ) • be a morphism of groupoids (Proposition 1.2), and let [ f /u ] : [ X/G ] → [ Y /H ] be the associated morphism of stacks. For ( P, a ) ∈ [ X/G ]( U ) , [ f /u ]( P, a ) is the pair consistingof the H -torsor P ∧ G,v H (where v is the composition of a × id G : P × G → X × G and u )and the H -equivariant morphism a v : P ∧ G,v H → Y defined by a via (1.5.5) .(b) Let ϕ , ϕ , r be as in Proposition 1.2 (b). Then the 2-morphism [ r ] : [ f /u ] → [ f /u ] inducedby r is given by the Y -morphism P ∧ G,v H → P ∧ G,v H sending ( p, h ) to ( p, r ( a ( p )) − h ) . For a crossed homomorphism u : X × G → H , the unit section of H defines a u -equivariantmorphism(1.6.1) X → X ∧ G,u H. The morphism of E -stacks(1.6.2) [ X/G ] → [( X ∧ G,u H ) /H ]induced by (1.6.1) sends T → X to T ∧ G,u H → X ∧ G,u H .6 emark 1.7. The restriction of (1.4.3) to Grpd Eq ( E ) is not fully faithful in general. In otherwords, for objects ( X, G ), (
Y, H ) of Eq( E ), a morphism of stacks [ X/G ] → [ Y /H ] does not neces-sarily come from a morphism of groupoids (
X, G ) • → ( Y, H ) • . In fact, in the case G = { } and Y is a nontrivial H -torsor over X , any quasi-inverse of the equivalence [ Y /H ] → X does not comefrom a morphism of groupoids. See Proposition 1.19 for a useful criterion. See also [47, Proposition5.1] for a calculus of fractions for the composite functor Eq( E ) → Stack( E ) of (1.1.1) and (1.4.3). Definition 1.8.
We say that a morphism X → Y in a 2-category C is faithful (resp. a monomor-phism ) if for every object U of C , the functor Hom( U, X ) → Hom(
U, Y ) is faithful (resp. fullyfaithful).In a 2-category, we need to distinguish between 2-limits [18, Definition 1.4.26] and strict 2-limits (called “2-limits” in [4, Definition 7.4.1]). Strict 2-products are 2-products. If a diagram X → Y ← X ′ in C admits a 2-fiber product X × Y X ′ and a strict 2-fiber product Z , the canonicalmorphism Z → X × Y X ′ is a monomorphism.In a (2,1)-category C admitting 2-fiber products, a morphism X → Y is faithful (resp. amonomorphism) if and only if its diagonal morphism X → X × Y X is a monomorphism (resp. anequivalence).A morphism of E -prestacks X → Y is faithful (resp. a monomorphism, resp. an equivalence) ifand only if X ( U ) → Y ( U ) is a faithful functor (resp. a fully faithful functor, resp. an equivalenceof categories) for every U in E . If X ′ is an E -prestack and X is its associated E -stack, then thecanonical morphism X ′ → X is a monomorphism.Let ( f, u ) : ( X, G ) → ( Y, H ) be a morphism of Eq( E ). If u is a monomorphism, then [ f /u ] : [ X/G ] → [ Y /H ] is faithful.
Remark 1.9.
The category Eq( C ) admits finite limits, whose formation commutes with theprojection functors ( X, G ) X and ( X, G ) G from Eq( C ) to C and to the category of groupobjects of C , respectively. The 2-category Grpd( C ) admits finite strict 2-limits, whose formationcommutes with the projection 2-functors X • X and X • X from Grpd( C ) to C . The functorEq( C ) → Grpd( C ) (1.1.1) sending ( X, G ) to (
X, G ) • carries finite limits to finite strict C ) admits finite 2-limits as well. The 2-fiber product of a diagram X • f −→ Y • g ←− Y ′• in Grpd( C ) is the groupoid W • of triples ( x, y, α ), where x ∈ X , y ∈ Y ′ , and( α : f ( x ) ∼ −→ g ( y )) ∈ Y . More formally, W = X × Y ,s Y Y × t Y ,Y Y ′ and W is the limit of thediagram X → Y p ←− Y × Y Y m −→ Y m ←− Y × Y Y p −→ Y ← Y ′ . A morphism X • → Y • in Grpd( C ) is faithful (resp. a monomorphism) if and only if the morphism X → ( X × X ) × Y × Y , ( s Y ,t Y ) Y is a monomorphism (resp. isomorphism).The category Stack( E ) admits small 2-limits. The pseudofunctor Grpd( E ) → Stack( E ) (1.4.3)preserves finite 2-limits and thus preserves faithful morphisms and monomorphisms. Remark 1.10.
A commutative square in Eq( E ),(1.10.1) ( X ′ , G ′ ) ( f ′ ,γ ′ ) / / ( p,u ) (cid:15) (cid:15) ( Y ′ , H ′ ) ( q,v ) (cid:15) (cid:15) ( X, G ) ( f,γ ) / / ( Y, H )induces a 2-commutative square of E -stacks(1.10.2) [ X ′ /G ′ ] / / (cid:15) (cid:15) [ Y ′ /H ′ ] (cid:15) (cid:15) [ X/G ] / / [ Y /H ] . It is not true in general that if (1.10.1) is cartesian, (1.10.2) is 2-cartesian, as (1.5.3) already shows.However, we have the following result, which is a partial generalization of [47, Proposition 5.4].7 roposition 1.11.
Consider a cartesian square (1.10.1) in Eq( E ) . If the morphism (1.11.1) H ′ × G → H in E given by ( h, g ) v ( h ) γ ( g ) is an epimorphism, then (1.10.2) is 2-cartesian.Proof. Let α : [ X ′ /G ′ ] → X := [ X/G ] × [ Y/H ] [ Y ′ /H ′ ]be the induced morphism of E -stacks. By Remark 1.9 and the remark following Definition 1.8, α is amonomorphism. We need to show that for any object V of E , the functor α V : [ X ′ /G ′ ] V → X V is es-sentially surjective. By definition, X V is the category of triples (( T, t ) , ( T ′ , t ′ ) , s ), where ( T, t : T → X ) is an object of [ X/G ] V , ( T ′ , t ′ : T ′ → Y ′ ) is an object of [ Y ′ /H ′ ] V , and s : [ f /u ] V ( T, t ) → [ q/v ] V ( T ′ , t ′ ) is an isomorphism. In other words (Proposition 1.6 (b)), s : T ∧ G,γ H → T ′ ∧ H ′ ,v H is an isomorphism of H -torsors over V , compatible with the morphisms to Y (induced by qt ′ and f t ). The functor α V sends an object ( P, w ) of [ X ′ /G ′ ] V to ([ p/u ] V ( P, w ) , [ f ′ /γ ′ ] V ( P, w ) , σ ), where σ : [ f p/γu ] V ( P, w ) → [ qf ′ /vγ ′ ] V ( P, w ) is the obvious isomorphism. Let a = (( T, t ) , ( T ′ , t ′ ) , s ) bean object of X V . It remains to show that there exist a cover ( V i → V ) i ∈ I and, for every i ∈ I ,an object ( P i , w i ) of [ X ′ /G ′ ] V i such that α ( P i , w i ) ≃ a V i . Take a cover ( V i → V ) i ∈ I such that forevery i , T V i and T ′ V i are both trivial and choose trivializations of them. Then s V i is representedby the left multiplication by some h i ∈ H ( V i ). By the assumption on (1.11.1), we may assume h i = v ( h ′ i ) γ ( g i ), h ′ i ∈ H ′ ( V i ), g i ∈ G ( V i ). In this case, the square(1.11.2) H V i s Vi / / λ γ ( gi ) (cid:15) (cid:15) H V i λ − v ( h ′ i ) (cid:15) (cid:15) H V i / / H V i commutes, where λ h is the left multiplication by h . Thus (1.11.2) gives an isomorphism a V i ≃ b i ,where b i = (( G V i , tλ − g i ) , ( H ′ V i , t ′ λ h ′ i ) , G V i , tλ − g i ) and ( H ′ V i , t ′ λ h ′ i ) over( H V i , ( tλ − g i ) γ = ( t ′ λ h ′ i ) v ) gives us an element ( P i , w i ) of [ X/G ] V i whose image under α is b i . Corollary 1.12.
Suppose u : G → Q is an epimorphism of groups of E , with kernel K . Then thenatural morphism BK ∼ −→ e × BQ BG is an equivalence. In other words, we can view Bu : BG → BQ as a fibration of fiber BK . Definition 1.13.
We say that a groupoid X • in E is an equivalence relation if ( s X , t X ) : X → X × X is a monomorphism. In this case, the associated E -stack [ X • ] is represented by thequotient sheaf in E . We say that the action of G on X is free if the associated groupoid ( X, G ) • is an equivalence relation. In this case, [ X/G ] is represented by the sheaf
X/G . Proposition 1.14.
Let ( X, G ) be an object in Eq( E ) , and let K be a normal subgroup of G actingfreely on X . Then the morphism f : [ X/G ] → [( X/K ) / ( G/K )] is an equivalence.Proof. Indeed, for every U in E , [( X/K ) / ( G/K )] U is the category of pairs ( T, α ), where T is a G/K -torsor and α : T → X/K is a G -equivariant map, and the functor f U admits a quasi-inversecarrying ( T, α ) to its base change by the projection X → X/K .The following induction formula will be useful later in the calculation of equivariant cohomologygroups (cf. [36, (1.7)]).
Corollary 1.15.
Let ( X, G ) be an object of Eq( E ) and let u : X × G → H be a crossed homomor-phism. Assume that the action of G on X × H , as defined in Notation 1.5, is free (Definition 1.8).Then f : [ X/G ] → [ X ∧ G,u
H/H ] (1.6.2) is an equivalence. roof. The morphism f can be decomposed as[ X/G ] α −→ [ X × H/G × H ] β −→ [ X ∧ G,u
H/H ] , where β is an equivalence by Proposition 1.14, and α is induced by the morphism X → X × H given by the unit section of H and the crossed homomorphism X × G → G × H sending ( x, g )to ( g, u ( x, g )). Since α is a 2-section of the morphism [ X × H/G × H ] → [ X/G ], which is anequivalence by Proposition 1.14, α is also an equivalence. Corollary 1.16.
Let u : H ֒ → G be a monomorphism of group objects in E . Then(a) The morphism of stacks BH → [( H \ G ) /G ] is an equivalence.(b) The natural morphism H \ G → e × BG BH is an isomorphism. In other words, (a) says that, for any homogeneous space X of group G , if H is the stabilizerof a section x of X , then the morphism BH → [ X/G ] given by x : e → X is an equivalence, while(b) can be thought as saying that BH → BG is a fibration of fiber H \ G . Proof.
Assertion (a) follows from Corollary 1.15. Assertion (b) follows from Proposition 1.11applied to the cartesian square ( H \ G, { } ) / / (cid:15) (cid:15) ( e, { } ) (cid:15) (cid:15) ( H \ G, G ) / / ( e, G )(cf. the paragraph following (1.5.3)) and from (a). Construction 1.17.
We will apply the above formalism to a relative situation, which we nowdescribe. Let X be an E -stack. We denote by Stack / X the (2,1)-category of E -stacks over X . Anobject of Stack / X is a pair ( Y , y ), where Y is an E -stack and y : Y → X is a morphism of E -stacks.A morphism in Stack / X from ( Y , y ) to ( Z , z ) is a pair ( f, α ), where f : Y → Z is a morphism of E -stacks and α : y → zf is a 2-morphism:(1.17.1) Y f / / y ❆❆❆❆❆❆❆ Z z (cid:15) (cid:15) X . ✁✁✁✁ < D A 2-morphism ( f, α ) → ( g, β ) in Stack / X is a 2-morphism η : f → g in the (2,1)-category Stack( E )such that β = ( z ∗ η ) ◦ α .A morphism y : Y → X of E -stacks is faithful (Definition 1.8) if and only if for any object U of E and any morphism x : U → X , the 2-fiber product U × x, X ,y Y is isomorphic to a sheaf. Considerthe 2-subcategory S of Stack / X spanned by objects ( Y , y ) with y faithful. For any morphism( f, α ) : ( Y , y ) → ( Z , z ) in S , f is necessarily faithful. A 2-morphism η : ( f, α ) → ( g, β ) in S , if itexists, is uniquely determined by ( f, α ) and ( g, β ). In other words, if we denote by Stack faith / X thecategory obtained from S by identifying isomorphic morphisms, then the 2-functor S →
Stack faith / X is a 2-equivalence.For any morphism φ : X → Y of E -stacks, base change by φ induces a functor Stack faith / Y → Stack faith / X . If S is an object of E , Stack faith /S is equivalent to E /S . More generally, if U • is a groupoidin E , Stack faith / [ U • ] is equivalent to the category of descent data relative to U • . In particular, if ( X, G )is an object of Eq( E ), Stack faith / [ X/G ] is equivalent to the category of G -objects of E , equivariant over X . For example, Stack faith /BG is equivalent to the topos B G of Grothendieck. Proposition 1.18. (a) The category
Stack faith / X is a U -topos. b) Let X be a stack. For any stack Y over X , associating to any stack Z faithful over X thegroupoid Hom X ( Z , Y ) defines a stack Y over Stack faith / X . The 2-functor (1.18.1) Stack / X → Stack(Stack faith / X ) , Y 7→ Y is a 2-equivalence.Proof. (a) We apply Giraud’s criterion [50, IV Théorème 1.2]. If T is a small generating familyof E , then ` U ∈T Ob( X ( U )) is an essentially small generating family of Stack faith / X . Let us nowshow that every sheaf F on Stack faith / X for the canonical topology is representable. Consider, forevery object U of E , the category of pairs ( x, s ) consisting of x ∈ X ( U ) and s ∈ Γ( x, F ), wherethe last occurrence of x is to be understood as the object x : U → X in Stack faith / X . A morphism( x, s ) ( y, t ) is a morphism α : x → y in X ( U ) such that α ∗ t = s . This defines an E -stack X ′ .The faithful morphism X ′ → X of E -stacks defined by the first projection ( x, s ) x represents F . The other conditions in Giraud’s criterion are trivially satisfied. Thus Stack faith / X is a U -topos.(b) We construct a 2-quasi-inverse to (1.18.1) as follows. Let C be a stack over Stack faith / X . Forevery object U of E , consider the category of pairs ( x, s ) consisting of x ∈ X ( U ) and s ∈ C ( x ).A morphism ( x, s ) → ( y, t ) is a pair ( α, β ) consisting of a morphism α : x → y in X ( U ) and amorphism β : α ∗ t → s in C ( x ). This defines an E -stack Y . The first projection ( x, s ) x definesa morphism Y → X of E -stacks. The construction C 7→ ( Y → X ) defines a pseudofunctor(1.18.2) Stack(Stack faith / X ) → Stack / X , which is a 2-quasi-inverse to (1.18.1).The composition of (1.4.3) and (1.18.2) is a faithful and essentially surjective (Definition 1.3)pseudofunctor(1.18.3) Grpd(Stack faith / X ) → Stack / X . We denote the image of a groupoid X • in Stack faith / X under (1.18.3) by [ X • / X ], and the image of amorphism f • of groupoids under (1.18.3) by [ f • / X ]. For ( X, G ) in Eq(Stack faith / X ), we denote theimage of ( X, G ) • under (1.18.3) by [ X/G/ X ]. For ( f, u ) : ( X, G ) • → ( Y, H ) • , we denote the imageunder (1.18.3) by [ f /u/ X ].We now apply the above formalism to the big fppf topoi of algebraic spaces. Recall that a stackis a stack over the big fppf site of Spec Z . The following result will be useful in Sections 7 and 8. Proposition 1.19.
Let X be a stack, and let X • , Y • be objects in Grpd(Stack faith / X ) . Assume that X is a strictly local scheme and the morphisms Y ⇒ Y are representable and smooth. Then thefunctor induced by (1.18.3) : F : Hom Grpd(Stack faith / X ) ( X • , Y • ) → Hom
Stack / X ([ X • / X ] , [ Y • / X ]) is an equivalence of categories.Proof. It remains to show that F is essentially surjective. Let φ : [ X • / X ] → [ Y • / X ] be a morphismin Stack / X . For the 2-cartesian square X ′ / / (cid:15) (cid:15) Y (cid:15) (cid:15) X / / [ X • / X ] φ / / [ Y • / X ] . Since X ′ is representable and smooth over X , it admits a section by [22, Corollaire 17.16.3 (ii),Proposition 18.8.1], which induces a 2-commutative square X (cid:15) (cid:15) f / / Y (cid:15) (cid:15) [ X • / X ] φ / / [ Y • / X ] . f = f × φ f : X → Y . Then f • : X • → Y • is a morphism of groupoids in Stack faith / X and φ ≃ [ f • / X ]. Remark 1.20.
Let X be a stack. We denote by Stack rep / X the full subcategory of Stack faith / X con-sisting of representable morphisms X → X . A morphism in this category from X → X to Y → X is an isomorphism class of pairs ( f, α ) (1.17.1). The morphisms f : X → Y are necessarily repre-sentable. Assume that X is an Artin stack. For any object X → X of Stack rep / X , X is necessarilyan Artin stack. For any object X • in Grpd(Stack rep / X ), if s X and t X are flat and locally of finitepresentation, then [ X • / X ] is an Artin stack. In particular, for any object ( X, G ) in Eq(Stack rep / X )with G flat of and locally of finite presentation over X , [ X/G/ X ] is an Artin stack. Notation 2.1.
Let X be an Artin stack. We denote by AlgSp / X the full subcategory of Stack rep / X (Remark 1.20) consisting of morphisms U → X with U an algebraic space. We let Sp sm / X denotethe full subcategory of AlgSp / X spanned by smooth morphisms U → X . The covering families ofthe smooth pretopology on Sp sm / X are those ( U i → U ) i ∈ I such that ` i ∈ I U i → U is smooth andsurjective. The covering families for the étale pretopology on Sp sm / X are those ( U i → U ) i ∈ I such that ` i ∈ I U i → U is étale and surjective. Since every smooth cover in Sp sm / X has an étale refinement by[22, Corollaire 17.16.3 (ii)], the smooth pretopology and the étale pretopology generate the sametopology on Sp sm / X (cf. [31, Définition 12.1]). We let X sm denote the associated topos, and call itthe smooth topos of X . Notation 2.2.
The category of sheaves in X sm is equivalent to the category of systems ( F u , θ φ ),where u : U → X runs through objects of Sp sm / X , φ : u → v runs through morphisms of Sp sm / X , F u isan étale sheaf on U , and θ φ : φ ∗ F v → F u , satisfying a cocycle condition [31, 12.2] and such that θ φ is an isomorphism for φ étale. Following [31, Définition 12.3], we say that a sheaf F on X is cartesian if θ φ is an isomorphism for all φ , or, equivalently, for all φ smooth (cf. [34, Lemma 3.8]).We denote by Sh cart ( X ) the full subcategory of Sh( X sm ) consisting of cartesian sheaves.Let Λ be a commutative ring. Following [31, Définition 18.1.4], we say, if Λ is noetherian,that a sheaf F of Λ-modules on X is constructible if F is cartesian and if F u is constructible forsome smooth atlas u : U → X , or equivalently, for every smooth atlas u : U → X . We denoteby Mod cart ( X , Λ) (resp. Mod c ( X , Λ)) the full subcategory of Mod( X sm , Λ) consisting of cartesian(resp. constructible) sheaves.We denote by D cart ( X , Λ) (resp. D c ( X , Λ)) the full subcategory of D ( X sm , Λ) consisting of com-plexes with cartesian (resp. constructible) cohomology sheaves. We have D c ( X , Λ) ⊂ D cart ( X , Λ).We will work exclusively with D cart ( X , Λ) rather than D ( X sm , Λ). We have functors ⊗ L Λ : D cart ( X , Λ) × D cart ( X , Λ) → D cart ( X , Λ) , R H om : D cart ( X , Λ) op × D cart ( X , Λ) → D cart ( X , Λ)defined on unbounded derived categories.If X is a Deligne-Mumford stack, we denote by X et or simply X its étale topos. The inclusionof the étale site in the smooth site induces a morphism of topoi ( ǫ ∗ , ǫ ∗ ) : X sm → X et . Note that ǫ ∗ is exact and ǫ ∗ induces an equivalence from X et to Sh cart ( X sm ). For any commutative ring Λ, ǫ ∗ induces D ( X , Λ) ∼ −→ D cart ( X , Λ).
Notation 2.3.
Let f : X → Y be a morphism of Artin stacks and let Λ be a commutative ring.Although the smooth topos is not functorial, we have a pair of adjoint functors f ∗ : Sh cart ( Y ) → Sh cart ( X ) , f ∗ : Sh cart ( X ) → Sh cart ( Y ) . and a pair of adjoint functors [32] f ∗ : D cart ( Y , Λ) → D cart ( X , Λ) , Rf ∗ : D cart ( X , Λ) → D cart ( Y , Λ) , where f ∗ is t -exact and Rf ∗ is left t -exact for the canonical t -structures. Note that Rf ∗ is definedon the whole category D cart , not just on D +cart . For M, N ∈ D cart ( Y , Λ), we have a naturalisomorphism f ∗ ( M ⊗ L Λ N ) ∼ −→ f ∗ M ⊗ L Λ f ∗ N. f is a surjective morphism, then the functors f ∗ are conservative and the functor f ∗ : Sh cart ( Y ) → Sh cart ( X ) is faithful.A 2-morphism α : f → g of morphisms of Artin stacks X → Y induces natural isomorphisms α ∗ : g ∗ → f ∗ and Rα ∗ : Rf ∗ → Rg ∗ . The following squares commute D cart ( Y , Λ) / / (cid:15) (cid:15) Rf ∗ f ∗ Rα ∗ (cid:15) (cid:15) g ∗ Rf ∗ α ∗ (cid:15) (cid:15) Rα ∗ / / g ∗ Rg ∗ (cid:15) (cid:15) Rg ∗ g ∗ α ∗ / / Rg ∗ f ∗ f ∗ Rf ∗ / / D cart ( X , Λ) . Recall that a morphism of Artin stacks f : X → Y is universally submersive [44, 06U6] if forevery morphism of Artin stacks Y ′ → Y , the base change Y ′ × Y X → Y ′ is submersive (on theunderlying topological spaces). Proposition 2.4.
Let f : X → Y be a morphism of Artin stacks. Assume that f is universallysubmersive (resp. faithfully flat and locally of finite presentation). Then f is of descent (resp.effective descent) for cartesian sheaves. Here effective descent means f ∗ induces an equivalence Sh cart ( Y ) ∼ −→ DD( f ) to the category ofdescent data, whose objects are cartesian sheaves F on X endowed with an isomorphism p ∗ F → p ∗ F satisfying the cocycle condition, where p , p : X × Y X → X are the two projections.
Proof.
By general properties of descent [19, Proposition 6.25, Théorème 10.4] and the case ofschemes [50, VIII Proposition 9.1] (resp. [50, VIII Théorème 9.4]), it suffices to show that smoothatlases are of effective descent for cartesian sheaves. In other words we may assume f is smoothand X is an algebraic space. In this case, we construct a quasi-inverse F of Sh cart ( Y ) → DD( f )as follows. Let A be a descent datum for f . For every object u : U → Y of Sp sm / Y , A inducesa descent datum A u for étale sheaves for the base change f u : X × Y U → U of f by u , and wetake ( F A ) u to be the corresponding étale sheaf on U . For a morphism φ : u → v in Sp sm / Y , we take φ ∗ ( F A ) v → ( F A ) u to be the isomorphism induced by the isomorphism of descent data φ ∗ A v → A u for étale sheaves for f u . Corollary 2.5.
Let S be an algebraic space, let G be a flat group algebraic S -space locally of finitepresentation, and let X be an algebraic space over S , endowed with an action of G . Denote by α : G × S X → X the action and by p : G × S X → X the projection, and let f : X → [ X/G ] bethe canonical morphism. Then f ∗ induces an equivalence of categories from Mod cart ([ X/G ]) tothe category of pairs ( F , a ) , where F ∈
Sh( X ) and a : α ∗ F → p ∗ F is a map satisfying the usualcocycle condition. Such pairs are called G -equivariant sheaves on X . The cocycle condition implies that i ∗ a : F →F is the identity, where i : X → G × S X is the morphism induced by the unit section of G . Proof.
This follows from Proposition 2.4 and the fact that f is faithfully flat of finite presentation. Corollary 2.6.
Let S and G be as in Corollary 2.5. Assume that G has connected geometricfibers. Let f : S → BG be a morphism corresponding to a G -torsor T on S . Then the functor f ∗ : Sh cart ( BG ) → Sh( S ) , is an equivalence.Proof. By Proposition 2.4, since f is faithfully flat locally of finite presentation, f ∗ induces anequivalence of categories from Sh cart ( BG ) to the category of pairs ( F , a ), where F is a sheaf on S and a : p ∗ F → p ∗ F is a descent datum with respect to f . As S × f,BG,f S is the sheaf H on S of G -automorphisms of T , and p = p is the projection p : H → S , a corresponds to an action of H on F . This action is trivial. Indeed, this can be checked over geometric points s → S , so wemay assume that S is the spectrum of an algebraically closed field. In this case, H ≃ G . As p ∗ F is constant and G is connected, and as the restriction of a to the unit section is the identity, a isthe identity. 12 emark 2.7. Corollary 2.6 implies that f ∗ and f ∗ are quasi-inverse to each other and the naturaltransformations id Sh cart ( BG ) → f ∗ f ∗ , f ∗ f ∗ → id Sh( S ) are natural isomorphisms. Since f is a2-section of the projection π : BG → S , we get natural isomorphisms π ∗ ≃ π ∗ f ∗ f ∗ ≃ f ∗ , π ∗ ≃ f ∗ f ∗ π ∗ ≃ f ∗ . In particular, we have natural isomorphisms f ∗ π ∗ ≃ id and π ∗ f ∗ ≃ id. Lemma 2.8.
Let X be an Artin stack, let Λ be a commutative ring, and let I ⊂ Z be an interval.For M ∈ D cart ( X , Λ) , the following conditions are equivalent:(a) For every N ∈ Mod cart ( X , Λ) , H q ( M ⊗ L Λ N ) = 0 for all q ∈ Z − I .(b) For every finitely presented Λ -module N , H q ( M ⊗ L Λ N ) = 0 for all q ∈ Z − I .(c) For every geometric point i : x → X , i ∗ M as an element of D ( x, Λ) is of tor-amplitudecontained in I . If the conditions of the lemma are satisfied, we say M is of cartesian tor-amplitude containedin I . If M ∈ D cart ( X , Λ) has cartesian tor-amplitude contained in [ a, + ∞ ) and N ∈ D ≥ b cart ( X , Λ),then M ⊗ L Λ N is in D ≥ a + b cart ( X , Λ).
Proof.
Obviously (a) implies (b) and (b) implies (c). Since the family of functors i ∗ : D cart ( X , Λ) → D ( x, Λ) is conservative, where i runs through all geometric points of X , (c) implies (a). Proposition 2.9 (Projection formula) . Let f : X → Y be a morphism of Artin stacks and let Λ be a commutative ring. Let L ∈ D cart ( X , Λ) , and let K ∈ D cart ( Y , Λ) such that H q K is constantfor all q . Assume one of the following:(a) Λ is noetherian regular and K ∈ D + c , L ∈ D + .(b) Λ is noetherian and K ∈ D bc (Λ) has finite cartesian tor-amplitude.(c) Rf ∗ : D cart ( X , Λ) → D cart ( Y , Λ) has finite cohomological amplitude, Λ is noetherian, K ∈ D c ,and either K, L ∈ D − or L has finite cartesian tor-amplitude.(d) f is quasi-compact quasi-separated, Λ is annihilated by an integer invertible on Y , K ∈ D + , L ∈ D + , and either Λ is noetherian regular or K has finite cartesian tor-amplitude.(e) f is quasi-compact quasi-separated, Λ is annihilated by an integer invertible on Y , and Rf ∗ : D cart ( X , Λ) → D cart ( Y , Λ) has finite cohomological amplitude.Then the map K ⊗ L Λ Rf ∗ L → Rf ∗ ( f ∗ K ⊗ L Λ L ) induced by the composite map f ∗ ( K ⊗ L Λ Rf ∗ L ) ∼ −→ f ∗ K ⊗ L Λ f ∗ Rf ∗ L → f ∗ K ⊗ L Λ L is an isomorphism.Proof. In case (a), we may assume that K is a (constant) Λ-module and we are then in case (b).In case (b), we may assume that Λ is local and it then suffices to take a finite resolution of K byfinite projective Λ-modules. In the first case of (c), we may assume K is a constant Λ-module.It then suffices to take a resolution of K by finite free Λ-modules. In the second case of (c), wereduce to the first case of (c) using Corollary 2.10 below of the first case of (c). In the first caseof (d), we may assume K ∈ D bc and we are in the second case of (d). In the second case of (d),we may assume that K is a flat Λ-module, thus a filtered colimit of finite free Λ-modules. Since R q f ∗ commutes with filtered colimits, we are reduced to the trivial case where K is a finite freeΛ-module. In case (e), since Rf ∗ preserves small coproducts, we may assume that L ∈ D − and K is represented by a complex in C − (Λ) of flat Λ-modules. We may further assume that L ∈ D b and K is a flat Λ-module. We are thus reduced to the second case of (d). Corollary 2.10.
Let f : X → Y be a morphism of Artin stacks and let Λ be a noetherian com-mutative ring. Assume that the functor Rf ∗ : D cart ( X , Λ) → D cart ( Y , Λ) has finite cohomologicalamplitude. Then, for every L ∈ D − cart ( X , Λ) of cartesian tor-amplitude contained in [ a, + ∞ ) , Rf ∗ L has cartesian tor-amplitude contained in [ a, + ∞ ) .Proof. This follows immediately from the first case of Proposition 2.9 (c) and Lemma 2.8.13he following statement on generic constructibility and generic base change generalizes [34,Theorem 9.10].
Proposition 2.11.
Let Z be an Artin stack and let f : X → Y be a morphism of Artin stacks offinite type over Z . Let Λ be a noetherian commutative ring annihilated by an integer invertibleon Z , and let L ∈ D + c ( X , Λ) . Then for every integer i there exists a dense open substack Z ◦ of Z such that(a) The restriction of R i f ∗ L to Z ◦ × Z Y ⊂ Y is constructible.(b) R i f ∗ L is compatible with arbitrary base change of Artin stacks Z ′ → Z ◦ ⊂ Z .Proof. Recall first that for any 2-commutative diagram of Artin stacks of the form X ′′ h ′ / / f ′′ (cid:15) (cid:15) X ′ g ′ / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′′ h / / Y ′ g / / Y the following diagram commutes:(2.11.1) ( gh ) ∗ Rf ∗ L b gh / / ≃ (cid:15) (cid:15) Rf ′′∗ ( g ′ h ′ ) ∗ L ≃ (cid:15) (cid:15) h ∗ g ∗ Rf ∗ L h ∗ b g / / h ∗ Rf ′∗ g ′∗ L b h / / Rf ′′∗ h ′∗ g ′∗ L where b gh , b g , b h are base change maps.If Z is a scheme, then, as in [34, Theorem 9.10], cohomological descent and the case of schemes[11, Th. finitude 1.9] imply that there exists a dense open subscheme Z ◦ of Z such that (a) holdsand that R i f ∗ L is compatible with arbitrary base change of schemes Z ′ → Z ◦ ⊂ Z . This implies(b). In fact, for any base change of Artin stacks g : Z ′ → Z ◦ ⊂ Z , take a smooth atlas p : Z ′ → Z ′ where Z ′ is a scheme. Then b p is an isomorphism and b gp is an isomorphism by assumption. Itfollows that p ∗ b g and hence b g are isomorphisms.In the general case, let p : Z → Z be a smooth atlas. By the preceding case, there exists adense open subscheme Z ◦ ⊂ Z such that after forming the 2-commutative diagram with 2-cartesiansquares X Zp X (cid:15) (cid:15) f Z / / Y Zp Y (cid:15) (cid:15) / / Z p (cid:15) (cid:15) X f / / Y / / Z the restriction of R i f Z ∗ p ∗X L to Z ◦ × Z Y Z is constructible and that R i f Z ∗ p ∗X L commutes witharbitrary base change of Artin stacks W → Z ◦ ⊂ Z . We claim that Z ◦ = p ( Z ◦ ) satisfies (a) and(b). To see this, let p ◦ : Z ◦ → Z ◦ be the restriction of p . By definition p ◦ is surjective. Then (a)follows from the fact that p ◦∗Y ( R i f ∗ L |Z ◦ × Z Y ) ≃ R i f Z ∗ p ∗X L | Z ◦ × Z Y Z is constructible. For any base change of Artin stacks Z ′ → Z ◦ , form the following 2-cartesiansquare: Z ′ h / / p ′ (cid:15) (cid:15) Z ◦ p ◦ (cid:15) (cid:15) Z ′ g / / Z ◦ . By (2.11.1), b p ′ ( p ′∗ b g ) can be identified with b h ( h ∗ b p ◦ ). Since p ◦ and p ′ are smooth, b p ◦ and b p ′ areisomorphisms. By the construction of p ◦ , b h is an isomorphism. It follows that p ′∗ b g and hence b g are isomorphisms. 14 emark 2.12. For Z = BG , where G is an algebraic group over a field k , f : X → Y a quasi-compact and quasi-separated morphism of Artin stacks over Z , and Λ is a commutative ringannihilated by an integer invertible in k , the above proof combined with the remark following [11,Th. finitude 1.9] shows that Rf ∗ : D +cart ( X , Λ) → D +cart ( Y , Λ) commutes with arbitrary base changeof Artin stacks Z ′ → Z . Definition 3.1.
For us, a ⊗ - category is a symmetric monoidal category [33, Section VII.7], that is,a category T endowed with a bifunctor ⊗ : T ×T → T , a unit object and functorial isomorphisms a LMN : L ⊗ ( M ⊗ N ) → ( L ⊗ M ) ⊗ N,c MN : M ⊗ N → N ⊗ M,u M : M ⊗ → M, v M : ⊗ M → M, satisfying the axioms of loc. cit. . We define a pseudo-ring in T to be an object K of T endowedwith a morphism m : K ⊗ K → K such that the following associativity diagram commutes: K ⊗ ( K ⊗ K ) a KKK (cid:15) (cid:15) id K ⊗ m / / K ⊗ K m " " ❋❋❋❋❋❋❋❋❋ ( K ⊗ K ) ⊗ K m ⊗ id K / / K ⊗ K m / / K. A pseudo-ring (
K, m ) is called commutative if the following diagram commutes K ⊗ K m ●●●●●●●●● c KK (cid:15) (cid:15) K ⊗ K m / / K. A homomorphism of pseudo-rings ( K, m ) → ( K ′ , m ′ ) is a morphism f : K → K ′ of T such thatthe following diagram commutes: K ⊗ K m / / f ⊗ f (cid:15) (cid:15) K f (cid:15) (cid:15) K ′ ⊗ K ′ m ′ / / K ′ . We define a left ( K, m ) -pseudomodule to be an object M of T endowed with a morphism n : K ⊗ M → M such that the following diagram commutes K ⊗ ( K ⊗ M ) id K ⊗ n / / a KKM (cid:15) (cid:15) K ⊗ M n ●●●●●●●●● ( K ⊗ K ) ⊗ M m ⊗ id M / / K ⊗ M n / / M. A homomorphism of left ( K, m ) -pseudomodules ( M, n ) → ( M ′ , n ′ ) is a morphism h : M → M ′ of T such that the following diagram commutes K ⊗ M n / / id K ⊗ h (cid:15) (cid:15) M h (cid:15) (cid:15) K ⊗ M ′ n ′ / / M ′ . efinition 3.2. Let f : ( K, m ) → ( K ′ , m ′ ) be a homomorphism of pseudo-rings. We define a splitting of f to be a morphism n : K ′ ⊗ K → K , making K into a ( K ′ , m ′ )-pseudomodule andsuch that the following diagram commutes K ⊗ K m % % ❑❑❑❑❑❑❑❑❑❑❑ f ⊗ id K / / K ′ ⊗ K n (cid:15) (cid:15) id K ′ ⊗ f / / K ′ ⊗ K ′ m ′ (cid:15) (cid:15) K f / / K ′ . Definition 3.3.
We define a ring in T to be a pseudo-ring ( K, m ) in T endowed with a morphism e : → K such that the following diagrams commute: K ⊗ id K ⊗ e / / u K % % ❑❑❑❑❑❑❑❑❑❑ K ⊗ K m (cid:15) (cid:15) ⊗ K e ⊗ id K / / v K % % ❑❑❑❑❑❑❑❑❑❑ K ⊗ K m (cid:15) (cid:15) K K. (Thus a ring in our sense is a “monoid” in the terminology of [33, Section VII.3].) The unit endowed with u : ⊗ → and id : → is a commutative ring in T . A ring homomorphism ( K, m, e ) → ( K ′ , m ′ , e ′ ) is a homomorphism of pseudo-rings f : ( K, m ) → ( K ′ , m ′ ) such that thefollowing diagram commutes: e / / e ′ ❆❆❆❆❆❆❆❆ K f (cid:15) (cid:15) K ′ . A left ( K, m, e ) -module is a left ( K, m )-pseudomodule (
M, n ) such that the following diagramcommutes ⊗ M v M % % ❑❑❑❑❑❑❑❑❑❑ e ⊗ M / / K ⊗ M n (cid:15) (cid:15) M. A homomorphism of left ( K, m, e ) -modules ( M, n ) → ( M ′ , n ′ ) is a homomorphism between theunderlying left ( K, m )-pseudomodules.
Construction 3.4.
Let T = ( T , ⊗ , a, c, u, v ) and T ′ = ( T ′ , ⊗ , a ′ , c ′ , u ′ , v ′ ) be ⊗ -categories, andlet ω : T → T ′ be a functor. A left-lax ⊗ -structure on ω is a natural transformation of functors T × T → T ′ consisting of morphisms of T ′ o MN : ω ( M ⊗ N ) → ω ( M ) ⊗ ω ( N ) , such that the following diagrams commute: ω ( L ⊗ ( M ⊗ N )) o L,M ⊗ N / / ω ( a LMN ) (cid:15) (cid:15) ω ( L ) ⊗ ω ( M ⊗ N ) ω ( L ) ⊗ o MN / / ω ( L ) ⊗ ( ω ( M ) ⊗ ω ( N )) a ′ ω ( L ) ω ( M ) ω ( N ) (cid:15) (cid:15) ω (( L ⊗ M ) ⊗ N ) o L ⊗ M,N / / ω ( L ⊗ M ) ⊗ ω ( N ) o LM ⊗ ω ( N ) / / ( ω ( L ) ⊗ ω ( M )) ⊗ ω ( N ) ω ( M ⊗ N ) o MN / / ω ( c MN ) (cid:15) (cid:15) ω ( M ) ⊗ ω ( N ) c ′ ω ( M ) ω ( N ) (cid:15) (cid:15) ω ( N ⊗ M ) o NM / / ω ( N ) ⊗ ω ( M ) . A right-lax ⊗ -structure on ω is a left-lax ⊗ -structure on ω op : T op → T ′ op . It is given by functorialmorphisms t MN : ω ( M ) ⊗ ω ( N ) → ω ( M ⊗ N ) , o inverted and replaced by t commute. A ⊗ -structure on ω is a left-lax ⊗ -structure o such that o MN is an isomorphism for all M and N . In this case t MN = o − MN defines a right-lax ⊗ -structure. If t is a right-lax ⊗ -structure on ω and ( K, m ) is apseudo-ring in T , we endow ω ( K ) with the pseudo-ring structure ω ( K ) ⊗ ω ( K ) t KK −−−→ ω ( K ⊗ K ) ω ( m ) −−−→ ω ( K ) . If, moreover, (
M, n ) is a left (
K, m )-pseudomodule, we endow ω ( M ) with the left ω ( K )-pseudomodulestructure ω ( K ) ⊗ ω ( M ) t KM −−−→ ω ( K ⊗ M ) ω ( n ) −−−→ ω ( M ) . If (
K, m ) is commutative, then ω ( K ) is commutative. This construction sends homomorphismsof pseudo-rings to homomorphisms of pseudo-rings and homomorphisms of left pseudomodules tohomomorphisms of left pseudomodules.If ( ω, t ), ( ω ′ , t ′ ) are functors endowed with right-lax ⊗ -structures, we say that a natural trans-formation α : ω → ω ′ preserves the right-lax ⊗ -structures if the following diagram commutes ω ( M ) ⊗ ω ( N ) t MN / / α M ⊗ α N (cid:15) (cid:15) ω ( M ⊗ N ) α M ⊗ N (cid:15) (cid:15) ω ′ ( M ) ⊗ ω ′ ( N ) t ′ MN / / ω ′ ( M ⊗ N ) . In this case, for any pseudo-ring K in T , α K : ω ( K ) → ω ′ ( K ) is a homomorphism of pseudo-rings. Construction 3.5.
Now suppose that ω : T → T ′ admits a right adjoint τ : T ′ → T . For any left-lax ⊗ -structure o on ω , endow τ with the right-lax ⊗ -structure t such that t MN : τ ( M ) ⊗ τ ( N ) → τ ( M ⊗ N ) is adjoint to the composition ω ( τ ( M ) ⊗ τ ( N )) o τ ( M ) τ ( N ) −−−−−−→ ω ( τ ( M )) ⊗ ω ( τ ( N )) α M ⊗ α N −−−−−→ M ⊗ N, where α M : ω ( τ ( M )) → M , α N : ω ( τ ( N )) → N are adjunction morphisms. It is straightforwardto check that this construction defines a bijection from the set of left-lax ⊗ -structures on ω to theset of right-lax ⊗ -structures on τ .In the above construction, if o is a ⊗ -structure on ω , then the adjunction morphisms α : ωτ → id T ′ and β : id T → τ ω preserve the resulting right-lax ⊗ -structures. Construction 3.6.
This formalism has a unital variant. A left-lax unital ⊗ -structure on a functor ω : T → T ′ is a left-lax ⊗ -structure endowed with a morphism p : ω ( ) → ′ in T ′ such that thefollowing diagrams commute ω ( M ⊗ ) o M / / ω ( u M ) (cid:15) (cid:15) ω ( M ) ⊗ ω ( ) id ω ( M ) ⊗ p (cid:15) (cid:15) ω ( ⊗ M ) o M / / ω ( v M ) (cid:15) (cid:15) ω ( ) ⊗ ω ( M ) p ⊗ id ω ( M ) (cid:15) (cid:15) ω ( M ) ω ( M ) ⊗ ′ u ′ ω ( M ) o o ω ( M ) ′ ⊗ ω ( M ) v ′ ω ( M ) o o A right-lax unital ⊗ -structure is a left-lax unital ⊗ -structure on ω op : T op → T ′ op . It consists of aright-lax ⊗ -structure endowed with a morphism s : ′ → ω ( ) in T ′ such that the above diagrams,with arrows o inverted and replaced by t , arrows p inverted and replaced by s , commute. A unital ⊗ -structure is a left-lax unital ⊗ -structure ( o, p ) such that o is a ⊗ -structure and p is invertible.Constructions 3.4 and 3.5 can be carried over to the unital case.Let T be a ⊗ -category, and let C be a category. Then the category T C of functors C → T hasa natural ⊗ -structure. The constant functor T → T C defined by M ( M ) C has a natural unital ⊗ -structure. Construction 3.7.
Let X = ( X, O X ) be a commutatively ringed topos. Two ⊗ -categories willbe of interest to us: 17a) The (unbounded) derived category D ( X ) = D ( X, O X ), equipped with ⊗ L O X : D ( X ) × D ( X ) → D ( X ) [27, Theorem 18.6.4].(b) The category GrMod( X ) = GrMod( X, O X ) of graded O X -modules H = L n ∈ Z H n , with ⊗ given by ( H ⊗ K ) n = L i + j = n H i ⊗ O X K j , the isomorphism c : H ⊗ K → K ⊗ H being givenby the usual sign rule.The cohomology functor H ∗ : D ( X ) → GrMod( X )has a natural right-lax unital ⊗ -structure given by the canonical maps H ∗ L ⊗H ∗ M → H ∗ ( L ⊗ L M ).(This is a unital ⊗ -structure when O X is a constant field, which is the case we are mostly interestedin).Let f : X = ( X, O X ) → Y = ( Y, O Y ) be a morphism of commutatively ringed topoi. We endow f ∗ : GrMod( Y ) → GrMod( X ) with the unital ⊗ -structure defined by the functorial isomorphisms f ∗ ( M ⊗ N ) → f ∗ M ⊗ f ∗ N, f ∗ O Y → O X . We endow Lf ∗ : D ( Y ) → D ( X ) [27, Theorem 18.6.9] with the unital ⊗ -structure defined by thefunctorial isomorphisms Lf ∗ ( M ⊗ L N ) → Lf ∗ M ⊗ L f ∗ N, Lf ∗ O Y → O X . We endow the right adjoint functors f ∗ : GrMod( X ) → GrMod( Y ) and Rf ∗ : D ( X ) → D ( Y ) withthe induced right-lax unital ⊗ -structures. Construction 3.8.
Let X be an Artin stack, and let Λ be a commutative ring. We considerthe ⊗ -categories D cart ( X , Λ) and GrMod cart ( X , Λ), the category of graded cartesian sheaves ofΛ-modules.Let f : X → Y be a morphism of Artin stacks. As in Construction 3.7, we endow the functors f ∗ : GrMod cart ( Y , Λ) → GrMod cart ( X , Λ) and f ∗ : D cart ( Y , Λ) → D cart ( X , Λ) with the naturalunital ⊗ -structures. We endow the right adjoint functors f ∗ : GrMod cart ( Y , Λ) → GrMod cart ( X , Λ)and Rf ∗ : D cart ( X , Λ) → D cart ( Y , Λ) with the induced right-lax unital ⊗ -structures.Assume that Λ is annihilated by an integer n invertible on Y and f is locally of finite presen-tation. Then we have Rf ! : D cart ( Y , Λ) → D cart ( X , Λ). As in [11, Cycle (1.2.2.3)], for M and N in D cart ( Y , Λ), we have a morphism f ∗ M ⊗ L Rf ! N → Rf ! ( M ⊗ L N )given by the morphism Rf ! N → Rf ! R H om ( M, M ⊗ L N ) ≃ R H om ( f ∗ M, Rf ! ( M ⊗ L N )). For apseudo-ring ( L, m ) in D cart ( Y , Λ), we endow Rf ! L with the left f ∗ L -pseudomodule structure givenby the composition f ∗ L ⊗ L Rf ! L → Rf ! ( L ⊗ L L ) Rf ! m −−−−→ Rf ! L Assume moreover that f = i is a closed immersion. Then the right-lax ⊗ -structure on i ∗ = Ri ∗ is an isomorphism and its inverse is a ⊗ -structure consisting of a functorial isomorphism i ∗ ( M ⊗ L N ) → i ∗ M ⊗ L i ∗ N. We endow the right adjoint functor Ri ! of i ∗ with the induced right-lax ⊗ -structure. Note that theright unital ⊗ -structure on i ∗ is not invertible in general. For a pseudo-ring ( L, m ) in D cart ( Y , Λ),the above left i ∗ L -pseudomodule structure on Ri ! L is a splitting of the homomorphism of pseudo-rings Ri ! L → i ∗ L (Definition 3.2).In the rest of this section, we discuss multiplicative structures on spectral objects. We will onlyconsider spectral objects of type ˜ Z , where ˜ Z is the category associated to the ordered set Z ∪{±∞} . Definition 3.9.
Let T be category endowed with a bifunctor ⊗ : T × T → T . Let J be a categoryendowed with a bifunctor ∗ : J × J → J . Let X, X ′ , X ′′ be functors J → T . A pairing from X , X ′ to X ′′ is a natural transformation of functors J × J → T consisting of morphisms of T X ( j ) ⊗ X ′ ( j ′ ) → X ′′ ( j ∗ j ′ ) . T , ⊗ ) and ( J, ∗ ) are endowed with structures of ⊗ -categories. A pairingfrom X , X to X is called associative if for j, j ′ , j ′′ ∈ J , the following diagram commutes X ( j ) ⊗ ( X ( j ′ ) ⊗ X ( j ′′ )) / / a (cid:15) (cid:15) X ( j ) ⊗ X ( j ′ ∗ j ′′ ) / / X ( j ∗ ( j ′ ∗ j ′′ )) a (cid:15) (cid:15) ( X ( j ) ⊗ X ( j ′ )) ⊗ X ( j ′′ ) / / X ( j ∗ j ′ ) ⊗ X ( j ′′ ) / / X (( j ∗ j ′ ) ∗ j ′′ ) , and is called commutative if for j, j ′ ∈ J , the following diagram commutes(3.9.1) X ( j ) ⊗ X ( j ′ ) / / c (cid:15) (cid:15) X ( j ∗ j ′ ) c (cid:15) (cid:15) X ( j ′ ) ⊗ X ( j ) / / X ( j ′ ∗ j ) . Assume moreover that T is additive and ⊗ is an additive bifunctor. Let S be the ⊗ -categorygiven by the discrete category {± } and the ordinary product. Let σ : J → S be a ⊗ -functor.A pairing from X , X to X is called σ -commutative if for j, j ′ ∈ J , the diagram (3.9.1) ismax { σ ( j ) , σ ( j ′ ) } -commutative. Construction 3.10.
Let Ar(˜ Z ) be the category of morphisms of ˜ Z = Z ∪ {±∞} . We representobjects of Ar(˜ Z ) by pairs ( p, q ), p, q ∈ ˜ Z , p ≤ q . We endow Ar(˜ Z ) with a structure of ⊗ -categoryby the formula ( p, q ) ∗ ( p ′ , q ′ ) = (max { p + q ′ − , p ′ + q − } , q + q ′ − . Here we adopt the convention that ( −∞ ) + (+ ∞ ) = −∞ = (+ ∞ ) + ( −∞ ). Definition 3.11.
Let D be a triangulated category endowed with a triangulated bifunctor ⊗ : D ×D → D [27, Definition 10.3.6]. Let (
X, δ ), ( X ′ , δ ), ( X ′′ , δ ′′ ) be spectral objects with values in D [46, II 4.1.2]. A pairing from ( X, δ ), ( X ′ , δ ′ ) to ( X ′′ , δ ′′ ) consists of a pairing from X , X ′ to X ′′ ,namely a natural transformation of functors Ar(˜ Z ) × Ar(˜ Z ) → D consisting of morphisms of D X ( p, q ) ⊗ X ′ ( p ′ , q ′ ) → X ′′ (( p, q ) ∗ ( p ′ , q ′ )) , such that for p ≤ q ≤ r , p ′ ≤ q ′ ≤ r ′ in ˜ Z satisfying q + r ′ = q ′ + r and p + r ′ = p ′ + r , the diagram X ( q, r ) ⊗ X ′ ( q ′ , r ′ ) / / ( δ ⊗ id , id ⊗ δ ′ ) (cid:15) (cid:15) X ′′ ( q ′′ , r ′′ ) δ ′′ (cid:15) (cid:15) ( X ( p, q )[1] ⊗ X ′ ( q ′ , r ′ )) ⊕ ( X ( q, r ) ⊗ X ′ ( p ′ , q ′ )[1]) / / X ′′ ( p ′′ , q ′′ )[1]commutes. Here ( q ′′ , r ′′ ) = ( q, r ) ∗ ( q ′ , r ′ ), ( p ′′ , q ′′ ) = ( p, q ) ∗ ( q ′ , r ′ ) = ( q, r ) ∗ ( p ′ , q ′ ).Assume moreover that ( D , ⊗ ) is endowed with a structure of ⊗ -category . A pairing from( X, δ ), (
X, δ ) to (
X, δ ) is called associative (resp. commutative ) if the underlying pairing from X , X to X is. Example 3.12.
Let X be a commutatively ringed topos, and let K, K ′ , K ′′ ∈ D ( X ). We considerthe second spectral object ( K, δ ) associated to K [46, III 4.3.1, 4.3.4], with K ( p, q ) = τ [ p,q − K ,where τ [ p,q − is the canonical truncation functor. Similarly, we have spectral objects ( K ′ , δ ′ ),( K ′′ , δ ′′ ). A map K ⊗ L K ′ → K ′′ in D ( X ) defines a pairing from ( K, δ ), ( K ′ , δ ′ ) to ( K ′′ , δ ′′ ) givenby τ [ p,q − K ⊗ L τ [ p ′ ,q ′ − K ′ → τ ≥ p ′′ ( τ [ p,q − K ⊗ L τ [ p ′ ,q ′ − K ′ ) ≃ τ ≥ p ′′ ( τ ≤ q − K ⊗ L τ ≤ q ′ − K ′ ) α −→ τ [ p ′′ ,q ′′ − ( K ⊗ L K ′ ) → τ [ p ′′ ,q ′′ − K ′′ , Here we do not assume that the constraints of the ⊗ -category are natural transformations of triangulatedfunctors [27, Definition 10.1.9 (ii)] in each variable. p ′′ , q ′′ ) = ( p, q ) ∗ ( p ′ , q ′ ), α is given by the map τ ≤ q − K ⊗ L τ ≤ q ′ − K ′ → τ ≤ q ′′ − ( K ⊗ L K ′′ )induced by adjunction from the map τ ≤ q − K ⊗ L τ ≤ q ′ − K ′ → K ⊗ L K ′ . Moreover, if K is apseudo-ring (resp. commutative pseudo-ring), then the induced pairing from ( K, δ ), (
K, δ ) to (
K, δ )is associative (resp. commutative).The above also holds with D ( X ) replaced by D cart ( X , Λ), where X is an Artin stack and Λ isa commutative ring. Definition 3.13.
Let A be an abelian category endowed with an additive bifunctor ⊗ : A×A → A .Let ( H n , δ ), ( H ′ n , δ ′ ), ( H ′′ n , δ ′′ ) be spectral objects with values in A [46, II 4.1.4]. A pairing from ( H n , δ ), ( H ′ n , δ ′ ) to ( H ′′ n , δ ′′ ) consists of a pairing from H ∗ , H ′∗ to H ′′∗ , namely a naturaltransformation of functors ( Z × Ar(˜ Z )) × ( Z × Ar(˜ Z )) → A consisting of morphisms of A H n ( p, q ) ⊗ H ′ n ′ ( p ′ , q ′ ) → H ′′ n + n ′ (( p, q ) ∗ ( p ′ , q ′ )) , such that for p ≤ q ≤ r , p ′ ≤ q ′ ≤ r ′ in ˜ Z satisfying q + r ′ = q ′ + r and p + r ′ = p ′ + r , the diagram H n ( q, r ) ⊗ H ′ n ′ ( q ′ , r ′ ) / / ( δ ⊗ id , ( − n id ⊗ δ ′ ) (cid:15) (cid:15) H ′′ n + n ′ ( q ′′ , r ′′ ) δ ′′ (cid:15) (cid:15) ( H n +1 ( p, q ) ⊗ H ′ n ′ ( q ′ , r ′ )) ⊕ ( H n ( q, r ) ⊗ H ′ n ′ +1 ( p ′ , q ′ )) / / H ′′ n + n ′ +1 ( p ′′ , q ′′ )commutes. Here ( q ′′ , r ′′ ) = ( q, r ) ∗ ( q ′ , r ′ ), ( p ′′ , q ′′ ) = ( p, q ) ∗ ( q ′ , r ′ ) = ( q, r ) ∗ ( p ′ , q ′ ). Note thatif ( H n , δ ), ( H ′ n , δ ′ ), and ( H ′′ n , δ ′′ ) are stationary [46, II 4.4.2], then the pairing from H ∗ , H ′∗ to H ′′∗ is uniquely determined by the pairing from H ∗ | Ar − , H ′∗ | Ar − to H ′′∗ | Ar − , whereAr − = Ar( Z ∪ {−∞} ). In fact, in this case, for every n , there exists an integer u ( n ) such that forevery q ≥ u ( n ), the morphism H n ( −∞ , q ) → H n ( −∞ , + ∞ ) is an isomorphism.Consider the induced spectral sequences ( E pq ⇒ H n ), ( E ′ pq ⇒ H ′ n ), ( E ′′ pq ⇒ H ′′ n ) givenby [46, II (4.3.3.2)]. A pairing from ( H n , δ ), ( H ′ n , δ ′ ) to ( H ′′ n , δ ′′ ) induces compatible pairings ofdifferential bigraded objects of A E pqr ⊗ E ′ p ′ q ′ r → E ′′ p + p ′ ,q + q ′ r for 2 ≤ r ≤ ∞ (satisfying d ′′ r ( xy ) = d r ( x ) y + ( − p + q xd ′ r ( y ) for x ∈ E pqr , y ∈ E ′ p ′ q ′ r ) and a pairingof filtered graded objects of A F p H n ⊗ F p ′ H ′ n ′ → F p + p ′ H ′′ n + n ′ , compatible with the pairing on E ∞ .Assume moreover that ( A , ⊗ ) is endowed with a structure of ⊗ -category. A pairing from ( H n , δ ),( H n , δ ) to ( H n , δ ) is called associative (resp. commutative ) if the underlying pairing from H ∗ , H ∗ to H ∗ is associative (resp. σ -commutative, where σ : Z × Ar(˜ Z ) → S is given by ( n, ( p, q )) ( − n ).An associative (resp. commutative) pairing from ( H n , δ ), ( H n , δ ) to ( H n , δ ) induces associative(resp. commutative) pairings on E pqr and F p H n . Here the commutativity for E pqr and F p H n arerelative to the functors Z × Z → S given by ( p, q ) ( − p + q and ( p, n ) ( − n , respectively. Remark 3.14.
Let D , D ′ be triangulated categories endowed with triangulated bifunctors ⊗ : D ×D → D , ⊗ : D ′ × D ′ → D ′ . Let τ : D → D ′ be a triangulated functor endowed with a natural trans-formation of functors D × D → D ′ consisting of morphisms τ ( M ) ⊗ τ ( N ) → τ ( M ⊗ N ) of D ′ that isa natural transformation of triangulated functors in each variable. Let ( X, δ ), ( X ′ , δ ′ ), ( X ′′ , δ ′′ ) bespectral objects with values in D . Then a pairing from ( X, δ ), ( X ′ , δ ′ ) to ( X ′′ , δ ′′ ) induces a pairingfrom τ ( X, δ ), τ ( X ′ , δ ′ ) to τ ( X ′′ , δ ′′ ). If ( D , ⊗ ), ( D ′ , ⊗ ) are endowed with structures of ⊗ -categoriesand τ is a right-lax ⊗ -functor (Construction 3.4), then an associative (resp. commutative) pair-ing from ( X, δ ), (
X, δ ) to (
X, δ ) induces an associative (resp. commutative) pairing from τ ( X, δ ), τ ( X, δ ) to τ ( X, δ ).Similarly, let A be an abelian category endowed with an additive bifunctor ⊗ : A × A → A and let H : D → A be a cohomological functor endowed with a natural transformation of functors For the filtration, we use the convention F p H n = Im( H n ( −∞ , n − p + 1) → H n ( −∞ , ∞ )). In particular, inExample 3.15 below, F p H n ( X, K ) = Im( H n ( X, τ ≤ n − p K ) → H n ( X, K )). × D → A consisting of morphisms H ( M ) ⊗ H ( N ) → H ( M ⊗ N ) of A . We adopt the conventionthat for p ≤ q ≤ r in ˜ Z , the map δ n : H n ( X ( q, r )) → H n +1 ( X ( p, q )) is ( − n times the mapobtained by applying H to δ [ n ] : X ( q, r )[ n ] → X ( p, q )[ n + 1]. Then a pairing from ( X, δ ), ( X ′ , δ ′ )to ( X ′′ , δ ′′ ) induces a pairing from H ∗ ( X, δ ), H ∗ ( X ′ , δ ′ ) to H ∗ ( X ′′ , δ ′′ ) given by H ( X ( p, q )[ n ]) ⊗ H ( X ′ ( p ′ , q ′ )[ n ′ ]) → H ( X ( p, q )[ n ] ⊗ X ′ ( p ′ , q ′ )[ n ′ ]) ≃ H (( X ( p, q ) ⊗ X ′ ( p ′ , q ′ ))[ n + n ′ ]) → H ( X ′′ (( p, q ) ∗ ( p ′ , q ′ ))[ n + n ′ ]) . Here we have used the composite of the isomorphisms M [ m ] ⊗ N [ n ] ≃ ( M ⊗ N [ n ])[ m ] ≃ ( M ⊗ N )[ m + n ]given by the structure of bifunctor of additive categories with translation [27, Definition 10.1.1(v)] on ⊗ : D × D → D . If ( D , ⊗ ), ( A , ⊗ ) are endowed with structures of ⊗ -categories and H is a right-lax ⊗ -functor, and if the associativity (resp. commutativity) constraint of ( D , ⊗ ) is anatural transformation of triangulated functors in each variable, then an associative (resp. com-mutative) pairing from ( X, δ ), (
X, δ ) to (
X, δ ) induces an associative (resp. commutative) pairingfrom H ∗ ( X, δ ), H ∗ ( X, δ ) to H ∗ ( X, δ ). Indeed, the assumption on the commutativity constraintimplies the ( − mn -commutativity of the following diagram M [ m ] ⊗ N [ n ] ∼ / / ≃ (cid:15) (cid:15) ( M ⊗ N [ n ])[ m ] ∼ / / ( M ⊗ N )[ m + n ] ≃ (cid:15) (cid:15) N [ n ] ⊗ M [ m ] ∼ / / ( N ⊗ M [ m ])[ n ] ∼ / / ( N ⊗ M )[ m + n ] . Example 3.15.
Let X be a commutatively ringed topos and let K be an object of D ( X ). Thesecond spectral sequence of hypercohomology E pq = H p ( X, H q K ) ⇒ H p + q ( X, K )is induced from the spectral object H ∗ ( K, δ ), where (
K, δ ) is the second spectral object associatedto K . If K is a pseudo-ring in D ( X ), then Remark 3.14 applied to Example 3.12 endows thespectral sequence with an associative multiplicative structure, which is graded commutative when K is commutative. Part II
Main results
We will first discuss Chern classes of vector bundles on Artin stacks. Let X be an Artin stack, let n ≥ X , and let L be a line bundle on X . The isomorphism class of L defines an element in H ( X , G m ). We denote by(4.0.1) c ( L ) ∈ H ( X , Z /n Z (1))the image of this element by the homomorphism H ( X , G m ) → H ( X , Z /n Z (1)) induced by theshort exact sequence 1 → Z /n Z (1) → G m n −→ G m → , where the map marked by n is raising to the n -th power. For any integer i , we write Z /n Z ( i ) = Z /n Z (1) ⊗ i . We say a quasi-coherent sheaf [44, 06WG] E on X is a vector bundle if there exists asmooth atlas p : X → X such that p ∗ E is a locally free O X -module of finite rank. The followingtheorem generalizes the construction of Chern classes of vector bundles on schemes ([39, Théorème1.3] and [51, VII 3.4, 3.5]). If X is a Deligne-Mumford stack, it yields the Chern classes over theétale topos of X locally ringed by O X , defined by Grothendieck in [21, (1.4)]. In particular, it alsogeneralizes [21, (2.3)]. 21 heorem 4.1. There exists a unique way to define, for every Artin stack X over Z [1 /n ] and everyvector bundle E on X , elements c i ( E ) ∈ H i ( X , Z /n Z ( i )) for all i ≥ such that the formal powerseries c t ( E ) = P i ≥ c i ( E ) t i satisfies the following conditions:(a) (Functoriality) If f : Y → X is a morphism of stacks over Z [1 /n ] , then f ∗ ( c t ( E )) = c t ( f ∗ E ) ;(b) (Additivity) If → E ′ → E → E ′′ → is an exact sequence of vector bundles, then c t ( E ) = c t ( E ′ ) c t ( E ′′ ) ;(c) (Normalization) If L is a line bundle on X , then c ( L ) coincides with the class definedin (4.0.1) and c t ( L ) = 1 X + c ( L ) t . Here X denotes the image of by the adjunctionhomomorphism Z /n Z → H ( X , Z /n Z ) .Moreover, we have:(d) c ( E ) = 1 X and c i ( E ) = 0 for i > rk( E ) . The c i ( E ) are called the (étale) Chern classes of E . It follows from (b) and (d) that c t ( E ) onlydepends on the isomorphism class of E .To prove Theorem 4.1, we need the following result, which generalizes [51, VII Théorème 2.2.1]and [39, Théorème 1.2]. Proposition 4.2.
Let X be an Artin stack and let E be a vector bundle of constant rank r on X . Let n be an integer invertible on X and let Λ be a commutative ring over Z /n Z . We denoteby π : P ( E ) → X the projective bundle of E . Let ξ = c ( O P ( E ) (1)) ∈ H ( P ( E ) , Λ(1)) as in (4.0.1) .Then the powers ξ i ∈ H i ( P ( E ) , Λ( i )) of ξ define an isomorphism in D ( X , Λ)(4.2.1) (1 , ξ, . . . , ξ r − ) : r − M i =0 Λ( − i )[ − i ] ∼ −→ Rπ ∗ Λ . Proof.
By base change [32], we reduce to the case of schemes, which is proven in [51, VII Théorème2.2.1].The uniqueness of Chern classes is a consequence of the following lemma, which generalizes [39,Propositions 1.4, 1.5].
Lemma 4.3.
Let X be an Artin stack, let n be an integer invertible on X , and let Λ be a commu-tative ring over Z /n Z .(a) (Splitting principle) Let E be a vector bundle on X of rank r and let π : F lag ( E ) → X be the fibration of complete flags of E . Then π ∗ E admits a canonical filtration by vectorbundles such that the graded pieces are line bundles, and the morphism Λ → Rπ ∗ Λ is a splitmonomorphism.(b) Let E : 0 → E ′ → E p −→ E ′′ → be a short exact sequence of vector bundles and let π : S ect ( E ) → X be the fibration of sections of p . Then S ect ( E ) is a torsor under H om ( E ′′ , E ′ ) and π ∗ E is canonically split. Moreover, the morphism Λ → Rπ ∗ Λ is an isomorphism.Proof. (a) follows from Proposition 4.2, as π is a composite of r successive projective bundles. For(b), up to replacing X by an atlas, we may assume that π is the projection from an affine space.In this case the assertion follows from [50, XV Corollaire 2.2].To define c i ( E ), we may assume E is of constant rank r . As usual, we define c i ( E ) ∈ H i ( X , Z /n Z ( i )) , ≤ i ≤ r, as the unique elements satisfying ξ r + X ≤ i ≤ r ( − i c i ( E ) ξ r − i = 0 , where ξ = c ( O P ( E ) (1)) ∈ H ( P ( E ) , Z /n Z (1)). We put c ( E ) = 1 and c i ( E ) = 0 for i > r . Theproperties (a) to (d) follow from the case of schemes. If c i ( E ) = 0 for all i >
0, in particular if E istrivial, then (4.2.1) is an isomorphism of rings in D ( X , Λ).22 heorem 4.4.
Let S be an algebraic space, let n be an integer invertible on S , and let Λ be acommutative ring over Z /n Z . Let N ≥ be an integer, let G = GL N,S , and let T = Q Ni =1 T i ⊂ G bethe subgroup of diagonal matrices, where T i = G m,S . Let π : BG → S , τ : BT → S , f ′ : G/T → S be the projections, let k : S → BG be the canonical section, let f : BT → BG be the morphisminduced by the inclusion T → G and let h : G/T → BT be the morphism induced by the projection G → S , as shown in the following 2-commutative diagram (4.4.1) G/T f ′ (cid:15) (cid:15) h / / BT f (cid:15) (cid:15) τ ! ! ❇❇❇❇❇❇❇❇ S k / / BG π / / S. Let E be the standard vector bundle of rank N on BG , corresponding to the natural representationof G in O NS . The i -th Chern class c i ( E ) of E induces a morphism α i : K i = Λ S ( − i )[ − i ] → Rπ ∗ Λ . Let L i be the inverse image on BT of the standard line bundle on BT i . Its first Chern class c ( L i ) induces a morphism β i : L i = Λ S ( − − → Rτ ∗ Λ . For a graded sheaf of Λ -modules M = L i ∈ Z M i on S , we let M ∆ = L i M i ( − i )[ − i ] ∈ D ( S, Λ) . Let Λ S [ x , . . . , x N ] (resp. Λ S [ t , . . . , t N ] ) be a polynomial algebra on generators x i of degree i (resp. t i ofdegree ). The corresponding object Λ S [ x , . . . , x N ] ∆ (resp. Λ S [ t , . . . , t N ] ∆ ) is naturally identifiedwith S Λ ( L ≤ i ≤ N K i ) (resp. S Λ ( L ≤ i ≤ N L i ) ). Then the ring homomorphisms α : Λ S [ x , . . . , x N ] ∆ → Rπ ∗ Λ , (4.4.2) β : Λ S [ t , . . . , t N ] ∆ → Rτ ∗ Λ , (4.4.3) defined respectively by α i and β i , are isomorphisms of rings in D ( S, Λ) , and fit into a commutativediagram of rings in D ( S, Λ)(4.4.4) Λ S [ x , . . . , x N ] ∆ σ / / α ≃ (cid:15) (cid:15) Λ S [ t , . . . , t N ] ∆ β ≃ (cid:15) (cid:15) ρ / / (Λ S [ t , . . . , t N ] / ( σ , . . . , σ N )) ∆ γ ≃ (cid:15) (cid:15) Rπ ∗ Λ a f / / Rτ ∗ Λ a h / / Rf ′∗ Λ , which commutes with arbitrary base change of algebraic spaces S ′ → S . Here σ sends x i to the i -thelementary symmetric polynomial σ i in t , . . . , t N , ρ is the projection, a f is induced by adjunctionby f and a h is induced by adjunction by h . Moreover, as graded module over R ∗ π ∗ Λ( ∗ ) , R ∗ τ ∗ Λ( ∗ ) is free of rank N ! . In particular, we have canonical decompositions Rπ ∗ Λ ≃ M q R q π ∗ Λ[ − q ] , Rτ ∗ Λ ≃ M q R q τ ∗ Λ[ − q ] , Rf ′∗ Λ ≃ M q R q f ′∗ Λ[ − q ] ,a h induces an epimorphism R ∗ τ ∗ Λ → R ∗ f ′∗ Λ and a f induces an isomorphism R ∗ π ∗ Λ ∼ −→ ( R ∗ τ ∗ Λ) S N ,where S N is the symmetric group on N letters. Moreover, (4.4.4) induces a commutative diagramof sheaves of Λ-algebras on S (4.4.5) Λ S [ x , . . . , x N ] σ / / α ≃ (cid:15) (cid:15) Λ S [ t , . . . , t N ] β ≃ (cid:15) (cid:15) ρ / / Λ S [ t , . . . , t N ] / ( σ , . . . , σ N ) γ ≃ (cid:15) (cid:15) R ∗ π ∗ Λ( ∗ ) a f / / R ∗ τ ∗ Λ( ∗ ) a h / / R ∗ f ′∗ Λ( ∗ ) , where α carries x i to the image of c i ( E ) under the edge homomorphism H i ( BG, Λ( i )) → H ( S, R i π ∗ Λ( i )) , β carries t i to the image of c ( L i ) under the edge homomorphism H i ( BT, Λ( i )) → H ( S, R i τ ∗ Λ( i )) . We will derive from Theorem 4.4 a formula for Rf ∗ (see Corollary 4.5). Proof.
As in [2, Lemma 2.3.1], we approximate BG by a finite Grassmannian G ( N, N ′ ) = M ∗ /G , where N ′ ≥ N , M is the algebraic S -space of N ′ × N matrices ( a ij ) ≤ i ≤ N ′ ≤ j ≤ N , M ∗ is the open subspaceof M consisting of matrices of rank N . Let B ⊂ G be the subgroup of upper triangular matrices.The square on the right of the diagram with 2-cartesian squares M ∗ y (cid:15) (cid:15) / / M ∗ /T p / / ψ (cid:15) (cid:15) w % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ M ∗ /B / / v (cid:27) (cid:27) ✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽ M ∗ /G φ (cid:15) (cid:15) u ~ ~ S g / / BT f / / ❴❴❴❴❴❴❴❴❴❴ τ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ BG π (cid:15) (cid:15) S induces a commutative square(4.4.6) Rπ ∗ Λ / / (cid:15) (cid:15) Rτ ∗ Λ (cid:15) (cid:15) Ru ∗ Λ / / Rv ∗ Λ a p / / Rw ∗ Λ . Here a p is induced by the adjunction Λ → Rp ∗ Λ. The latter is an isomorphism by [50, XVCorollaire 2.2], because p is a ( B/T )-torsor and
B/T is isomorphic to the unipotent radical of B ,which is an affine space over S . The diagram M − M ∗ i / / z $ $ ■■■■■■■■■■ M x (cid:15) (cid:15) M ∗ y } } ④④④④④④④④④ o o S induces an exact triangle Rz ∗ Ri ! Λ → Rx ∗ Λ → Ry ∗ Λ → . Since M is an affine space over S , the adjunction Λ → Rx ∗ Λ is an isomorphism [50, XV Corollaire2.2]. Since x is smooth and the fibers of z are of codimension N ′ − N + 1, we have Ri ! Λ ∈ D ≥ N ′ − N +1) by semi-purity [11, Cycle 2.2.8]. It follows that the adjunction Λ → τ ≤ N ′ − N ) Ry ∗ Λis an isomorphism. By smooth base change by g (resp. f g ) [32], this implies that the adjunctionΛ → τ ≤ N ′ − N ) Rψ ∗ Λ (resp. Λ → τ ≤ N ′ − N ) Rφ ∗ Λ) is an isomorphism, so that the right (resp. left)vertical arrow of τ ≤ N ′ − N ) (4.4.6) is an isomorphism.The assertions then follow from an explicit computation of Ru ∗ Λ and Rv ∗ Λ. Note that M ∗ /B isa partial flag variety of the free O S -module O N ′ S of type (1 , . . . , , N ′ − N ). By [51, VII Propositions5.2, 5.6 (a)] applied to u and v , we have a commutative square A ∆ ∼ / / σ (cid:15) (cid:15) Ru ∗ Λ (cid:15) (cid:15) C ∆ ∼ / / Rv ∗ Λ . This approximation argument was explained by Deligne to the first author in the context of de Rham cohomologyin 1967. A = Λ S [ x , . . . , x N , y , . . . , y N ′ − N ] / ( X i + j = m x i y j ) m ≥ ,C = Λ S [ t , . . . , t N , y , . . . , y N ′ − N ] / ( X i + j = m σ i y j ) m ≥ , the upper horizontal arrow sends x i to the i -th Chern class c i ( E N ′ ) of the canonical bundle E N ′ of rank N on the Grassmannian M ∗ /G , the lower horizontal arrow sends t i to the first Chernclass c ( L i,N ′ ) of the i -th standard line bundle L i,N ′ of the partial flag variety M ∗ /B , and theupper (resp. lower) horizontal arrow sends y i to the i -th Chern class c i ( E ′ N ′ ) of the canonicalbundle E ′ N ′ of rank N ′ − N on M ∗ /G (resp. on M ∗ /B ). In the definition of the ideals, we put x = y = 1, x i = 0 for i > N and y i = 0 for i > N ′ − N , and we used the fact that c m of thetrivial bundle of rank N ′ is zero for m ≥
1. As E N ′ (resp. L i,N ′ ) is induced from E (resp. L i ),by the functoriality of Chern classes (Theorem 4.1), these isomorphisms are compatible with themorphisms α (4.4.2) and β (4.4.3). We can rewrite A as Λ[ x , . . . , x N ] / ( P m ( x , . . . , x m )) m>N ′ − N and rewrite C as Λ[ t , . . . , t N ] / ( Q m ( t , . . . , t m )) m>N ′ − N , where P m is an isobaric polynomial ofweight m in x , . . . , x m , x i being of weight i , and Q m is a homogeneous polynomial of degree m in t , . . . , t m . As the vertical arrows of (4.4.6) induce isomorphisms after application of the truncationfunctor τ ≤ N ′ − N ) , it follows that τ ≤ N ′ − N ) of the square on the left of (4.4.4) is commutativeand the vertical arrows induce isomorphisms after application of τ ≤ N ′ − N ) . To get the square onthe right of (4.4.4), it suffices to apply the preceding computation of Rw ∗ Λ (via Rv ∗ Λ) to the case N ′ = N , because, in this case, f ′ = w . The fact that (4.4.4) commutes with base change followsfrom the functoriality of Chern classes. The last assertion of the theorem then follows from [51,VII Lemme 5.4.1]. Corollary 4.5.
With assumptions and notation as in Theorem 4.4:(a) For every locally constant Λ -module F on S , the projection formula maps F ⊗ L Λ Rπ ∗ Λ → Rπ ∗ π ∗ F , F ⊗ L Λ Rτ ∗ Λ → Rτ ∗ τ ∗ F are isomorphisms.(b) The classes c ( L i ) induce an isomorphism of rings Λ BG [ t , . . . , t N ] ∆ /J → Rf ∗ Λ in D ( BG, Λ) ,where J is the ideal generated by σ i − c i ( E ) . Moreover, the left square of (4.4.1) induces anisomorphism of Rπ ∗ Λ -modules Rτ ∗ Λ ≃ Rπ ∗ Λ ⊗ L Λ Rf ′∗ Λ .Proof. (a) We may assume that F is a constant Λ-module of value F . Then the assertion followsfrom Theorem 4.4 applied to Λ and to the ring of dual numbers Λ ⊕ F (with m m = 0 for m , m ∈ F ).(b) Since f ∗ E ≃ L Ni =1 L i , f ∗ c i ( E ) = c i ( f ∗ E ) is the i -th elementary symmetric polynomial in c ( L ) , . . . , c ( L N ). Thus the ring homomorphism Λ BG [ t , . . . , t N ] ∆ → Rf ∗ Λ induced by c ( L i )factorizes through a ring homomorphism Λ BG [ t , . . . , t N ] ∆ /J → Rf ∗ Λ. By Proposition 1.11 ap-plied to the square (
G, T ) / / (cid:15) (cid:15) ( S, T ) (cid:15) (cid:15) ( G, G ) / / ( S, G ) , the left square of (4.4.1) is 2-cartesian. By smooth base change by k , we have k ∗ Rf ∗ Λ ≃ Rf ′∗ Λ. Thefirst assertion then follows from Theorem 4.4. By Remark 2.7, it follows that Rf ∗ Λ ≃ π ∗ k ∗ Rf ∗ Λ ≃ π ∗ Rf ′∗ Λ. Thus Rτ ∗ Λ ≃ Rπ ∗ Rf ∗ Λ ≃ Rπ ∗ π ∗ Rf ′∗ Λ and the second assertion follows from (a).Let k be a separably closed field, let n be an integer invertible in k , and let Λ be a noetheriancommutative ring over Z /n Z . The next sequence of results are analogues of Quillen’s finitenesstheorem [36, Theorem 2.1, Corollaries 2.2, 2.3]. Recall that an algebraic space over Spec k is offinite presentation if and only if it is quasi-separated and of finite type.25 heorem 4.6. Let G be an algebraic group over k , let X be an algebraic space of finite presentationover Spec k equipped with an action of G , and let K be an object of D bc ([ X/G ] , Λ) (see Notation 2.2).Then H ∗ ( BG, Λ) is a finitely generated Λ -algebra and H ∗ ([ X/G ] , K ) is a finite H ∗ ( BG, Λ) -module.In particular, if K is a ring in the sense of Definition 3.3, then the graded center ZH ∗ ([ X/G ] , K ) of H ∗ ([ X/G ] , K ) is a finitely generated Λ -algebra. Initially the authors established Theorem 4.6 for G either a linear algebraic group or a semi-abelian variety. The finiteness of H ∗ ( BG,
Λ) in the general case was proved by Deligne in [12].
Corollary 4.7.
Let G be an algebraic group over k and let f : X → BG be a representable mor-phism of Artin stacks of finite presentation over Spec k , and let K ∈ D bc ( X , Λ) . Consider H ∗ ( X , K ) as an H ∗ ( BG, Λ) -module by restriction of scalars via the map f ∗ : H ∗ ( BG, Λ) → H ∗ ( X , Λ) . Then H ∗ ( X , K ) is a finite H ∗ ( BG, Λ) -module.Proof. It suffices to apply Theorem 4.6 to Rf ∗ K ∈ D bc ( BG,
Λ).
Corollary 4.8.
Let X (resp. Y ) be an algebraic space of finite presentation over Spec k , equippedwith an action of an algebraic group G (resp. H ) over k . Let ( f, u ) : ( X, G ) → ( Y, H ) be an equiv-ariant morphism. Assume that u is a monomorphism. Then the map [ f /u ] ∗ makes H ∗ ([ X/G ] , Λ) a finite H ∗ ([ Y /H ] , Λ) -module. Indeed, since the map [
X/G ] → BH induced by u is representable, H ∗ ([ X/G ] , Λ) is a finite H ∗ ( BH,
Λ)-module by Corollary 4.7, hence a finite H ∗ ([ Y /H ] , Λ)-module.
Proof of Theorem 4.6.
By the invariance of étale cohomology under schematic universal homeo-morphisms, we may assume k algebraically closed and G reduced (hence smooth). Then G is anextension 1 → G → G → F →
1, where F is the finite group π ( G ) and G is the identitycomponent of G . By Chevalley’s theorem (cf. [8, Theorem 1.1.1] or [9, Theorem 1.1]), G is anextension 1 → L → G → A →
1, where A is an abelian variety and L = G aff is the largestconnected affine normal subgroup of G . Then L is also normal in G , and if E = G/L , then E isan extension 1 → A → E → F →
1. We will sum up this dévissage by saying that G is an iteratedextension G = L · A · F .By [20, VIII 7.1.5, 7.3.7], for every algebraic group H over k , the extensions of F by H withgiven action of F on H by conjugation are classified by H ( BF, H ). In particular, the extension E of F by A defines an action of F on A and a class in H ( BF, A ), which comes from a class α in H ( BF, A [ m ]), where m is the order of F and A [ m ] denotes the kernel of m : A → A . Indeed thesecond arrow in the exact sequence H ( BF, A [ m ]) → H ( BF, A ) × m −−→ H ( BF, A )is equal to zero. This allows us to define an inductive system of subgroups E i = A [ mn i ] · F of E , given by the image of α in H ( F, A [ mn i ]). This induces an inductive system of subgroups G i = L · A [ mn i ] · F of G , fitting into short exact sequences1 / / L / / G i / / (cid:15) (cid:15) (cid:3) E i / / (cid:15) (cid:15) / / L / / G / / E / / . Form the diagram with cartesian squares[
X/G i ] / / f i (cid:15) (cid:15) BG i (cid:15) (cid:15) G/G i o o (cid:15) (cid:15) [ X/G ] / / BG Spec k. o o Note that
G/G i = A/A [ mn i ] and the vertical arrows in the above diagram are proper representable.By the classical projection formula [50, XVII (5.2.2.1)], Rf i ∗ f ∗ i K ≃ K ⊗ L Λ Rf i ∗ Λ. Moreover, f i ∗ Λ ≃ Λ. Thus we have a distinguished triangle(4.8.1) K → Rf i ∗ f ∗ i K → K ⊗ L Λ τ ≥ Rf i ∗ Λ → . N i = K ⊗ L Λ τ ≥ Rf i ∗ Λ forms an AR-nullsystem of level 2 d in the sense that N i +2 d → N i is zero for all i , where d = dim A . Indeed thestalks of R q f i ∗ Λ are H q ( A/A [ mn i ] , Λ), which is zero for q > d . For q = 0, the transition mapsof ( H ( A/A [ mn i ] , Λ)) are id Λ and for q >
0, the transition maps of ( H q ( A/A [ mn i ] , Λ)) are zero.Thus, in the induced long exact sequence of (4.8.1) H ∗− ([ X/G ] , N i ) → H ∗ ([ X/G ] , K ) α i −→ H ∗ ([ X/G i ] , f ∗ i K ) → H ∗ ([ X/G ] , N i ) , the system ( H ∗ ([ X/G ] , N i )) is AR-null of level 2 d . Therefore, α i is injective for i ≥ d andIm α i = Im( H ∗ ([ X/G i +2 d ] , f ∗ i +2 d K ) → H ∗ ([ X/G i ] , f ∗ i K )) for all i . Taking i = 2 d , we get H ∗ ([ X/G ] , K ) = Im( H ∗ ([ X/G d ] , f ∗ d K ) → H ∗ ([ X/G d ] , f ∗ d K )). In particular, H ∗ ( BG,
Λ) isa quotient Λ-algebra of H ∗ ( BG d , Λ), and H ∗ ([ X/G ] , K ) is a quotient H ∗ ( BG,
Λ)-module of H ∗ ([ X/G d ] , f ∗ d K ). Therefore, it suffices to show the theorem with G replaced by G d . In partic-ular, we may assume that G is a linear algebraic group.Let G → GL r be an embedding into a general linear group. By Corollary 1.15, the morphismof Artin stacks over B GL r , [ X/G ] → [( X ∧ G GL r ) / GL r ] , is an equivalence. Replacing G by GL r and X by X ∧ G GL r , we may assume that G = GL r .Let f : [ X/G ] → BG . Then Rf ∗ K ∈ D bc ( BG,
Λ). Thus we may assume X = Spec k . The fullsubcategory of objects K satisfying the theorem is a triangulated category. Thus we may furtherassume K ∈ Mod c ( BG,
Λ). In this case, since G is connected, K is necessarily constant (Corollary2.6) so that K ≃ π ∗ M for some finite Λ-module M , where π : BG → Spec k . In this case, byTheorem 4.4, H ∗ ( BG, Λ) ≃ Λ[ c , . . . , c r ] is a noetherian ring and H ∗ ( BG, K ) ≃ M ⊗ Λ Λ[ c , . . . , c r ]is a finite H ∗ ( BG,
Λ)-module.
Remark 4.9.
We have shown in the proof of Theorem 4.6 that H ∗ ([ X/G ] , K ) ≃ Im( H ∗ ([ X/G d ] , f ∗ d K ) → H ∗ ([ X/G d ] , f ∗ d K )). In particular, H ∗ ([ X/G ] , K ) is a quotient H ∗ ( BG,
Λ)-module of H ∗ ([ X/G d ] , f ∗ d K ).Here G d < G d are affine subgroups of G , independent of X and K , and f d : [ X/G d ] → [ X/G ], f d : [ X/G d ] → [ X/G ].In the following examples, we write H ∗ ( − ) for H ∗ ( − , Λ), with Λ as in Theorem 4.6.
Example 4.10.
Let
G/k be an extension of an abelian variety A of dimension g by a torus T ofdimension r . Then(a) H ( A ), H ( T ), H ( G ) are free over Λ of ranks 2 g , r and 2 g + r respectively, and the sequence0 → H ( A ) → H ( G ) → H ( T ) → H ( G ) ֒ → H ∗ ( G ) induces anisomorphism of Λ-modules(4.10.1) ∧ H ( G ) ∼ −→ H ∗ ( G ) . (b) The homomorphism d : H ( G ) → H ( BG )in the spectral sequence E pq = H p ( BG ) ⊗ H q ( G ) ⇒ H p + q (Spec k )of the fibration Spec k → BG is an isomorphism.(c) We have H i +1 ( BG ) = 0 for all i , and the inclusion H ( BG ) ֒ → H ∗ ( BG ) extends to anisomorphism of Λ-algebras S( H ( BG )) ∼ −→ H ∗ ( BG ) . Let us briefly sketch a proof.Assertion (a) is standard. By projection formula, we may assume Λ = Z /n Z . As the mul-tiplication by n on T is surjective, the sequence 0 → T [ n ] → G [ n ] → A [ n ] → π ( G ) → G [ n ] induces an injection Hom( G [ n ] , Z /n Z ) → H ( G ). The fact that thisinjection and (4.10.1) are isomorphisms follows (after reducing to n = ℓ prime) from the structureof Hopf algebra of H ∗ ( G ), as H g + r ( G ) ∼ −→ H r ( T ) ⊗ H g ( A ) is of rank 1 (cf. [42, Chapter VII,Proposition 16]). 27ssertion (b) follows immediately from (a).To prove (c) we calculate H ∗ ( BG ) using the nerve B • G of G (cf. [10, 6.1.5]): H ∗ ( BG ) = H ∗ ( B • G ) , which gives the Eilenberg-Moore spectral sequence:(4.10.2) E ij = H j ( B i G ) ⇒ H i + j ( BG ) . One finds that E • ,j ≃ L ∧ j ( H ( G )[ − . By [23, I 4.3.2.1 (i)] we get E • ,j ≃ L S j ( H ( G ))[ − j ] . Thus E ij ≃ ( S j ( H ( G )) if i = j ,0 if i = j .The E term is concentrated on the diagonal, hence (4.10.2) degenerates at E , and we get anisomorphism H ∗ ( BG ) = S( H ( G )[ − , from which (c) follows. Example 4.11.
Let G be a connected algebraic group over k . Assume that for every prime number ℓ dividing n , H i ( G, Z ℓ ) is torsion-free for all i . Classical results due to Borel [5] can be adapted asfollows.(a) H ∗ ( G ) is the exterior algebra over a free Λ-module having a basis of elements of odd degree[5, Propositions 7.2, 7.3].(b) In the spectral sequence of the fibration Spec k → BG , E ij = H i ( BG ) ⊗ H j ( G ) ⇒ H i + j (Spec k ) , primitive and transgressive elements coincide [5, Proposition 20.2], and the transgression givesan isomorphism d q +1 : P q ∼ −→ Q q +1 from the transgressive part P q = E qq +1 of H q ( G ) ≃ E q to the quotient Q q +1 = E q +1 , q +1 of H q +1 ( BG ) ≃ E q +1 , . Moreover, Q ∗ is a free Λ-modulehaving a basis of elements of even degrees, and every section of H ∗ ( BG ) → Q ∗ provides anisomorphism between H ∗ ( BG ) and the polynomial algebra S Λ ( Q ∗ ) [5, Théorèmes 13.1, 19.1].Now assume that G is a connected reductive group over k . Let T be the maximal torus in G ,and W = Norm G ( T ) /T the Weyl group. Recall that G is ℓ -torsion-free if ℓ does not divide theorder of W , cf. [6], [43, Section 1.3]. As in [11, Sommes trig., 8.2], the following results can bededuced from the classical results on compact Lie groups by lifting G to characteristic zero.(c) The spectral sequence(4.11.1) E ij = H i ( BG ) ⊗ H j ( G/T ) ⇒ H i + j ( BT )degenerates at E , E ij being zero if i or j is odd . In particular, the homomorphism H ∗ ( BT ) → H ∗ ( G/T )induced by the projection
G/T → BT is surjective. In other words, in view of Theorem 4.4, H ∗ ( G/T ) is generated by the Chern classes of the invertible sheaves L χ obtained by pushingout the T -torsor G over G/T by the characters χ : T → G m .(d) The Weyl group W acts on (4.11.1), trivially on H ∗ ( BG ), and H ∗ ( G/T ) is the regularrepresentation of W [5, Lemme 27.1]. In particular, the homomorphism H ∗ ( BG ) → H ∗ ( BT )induced by the projection BT → BG induces an isomorphism(4.11.2) H ∗ ( BG ) ∼ −→ H ∗ ( BT ) W . The vanishing of H j ( G/T ) for j odd follows for example from the Bruhat decomposition of G/B for a Borel B containing T . Finiteness of orbit types
Let k be a field of characteristic p ≥
0, let G be an algebraic group over k , and let A be a finitegroup. The presheaf of sets H om group ( A, G ) on AlgSp /k is represented by a closed subscheme X ofthe product Q a ∈ A G of copies of G indexed by A . In the case where A ≃ ( Z /ℓ Z ) r is an elementaryabelian ℓ -group of rank r , H om group ( A, G )( T ) can be identified with the set of commuting r -tuples of ℓ -torsion elements of G ( T ). The group G acts on X by conjugation. Let x ∈ X ( k )be a rational point of X and let c : G → X be the G -equivariant morphism sending g to xg ,where xg : a g − x ( a ) g . Let H = c − ( x ) ⊂ G be the inertia subgroup at x . The morphism c decomposes into G → H \ G f −→ X, where f is an immersion [13, III, § 3, Proposition 5.2]. The orbit of x under G is the (scheme-theoretic) image of f , which is a subscheme of X . The orbits of X are disjoint with each other.The following result is probably well known. It was communicated to us by Serre. Theorem 5.1 (Serre) . Assume that the order of A is not divisible by p . Then the orbits of X under the action of G are open subschemes. Moreover, if G is smooth, then X is smooth. The condition on the order of A is essential. For example, if p > A = Z /p Z and G = G a isthe additive group, then G acts trivially on X ≃ G .Note that for any field extension k ′ of k , if Y is an orbit of X under G , then Y k ′ is an orbit of X k ′ under G k ′ . Corollary 5.2.
The orbits are closed and the number of orbits is finite. Moreover, if k is alge-braically closed, then the orbits form a disjoint open covering of X .Proof. It suffices to consider the case when k is algebraically closed. In this case, rational pointsof X form a dense subset [22, Corollaire 10.4.8]. Thus, by Theorem 5.1, the orbits form a disjointopen covering of the quasi-compact topological space X . Therefore, the orbits are also closed andthe number of orbits is finite. Corollary 5.3.
Let G be an algebraic group over k and let ℓ be a prime number distinct from p .There are finitely many conjugacy classes of elementary abelian ℓ -subgroups of G . Moreover, if k is algebraically closed and k ′ is an algebraically closed extension of k , then the natural map S k → S k ′ from the set S k of conjugacy classes of elementary abelian ℓ -subgroups of G to the set S k ′ of conjugacy classes of elementary abelian ℓ -subgroups of G k ′ is a bijection.Proof. By Corollary 5.2, it suffices to show that the ranks of the elementary abelian ℓ -subgroupsof G are bounded. For this, we may assume k algebraically closed, and G smooth. As in the proofof Theorem 4.6, let L be the maximal connected affine normal subgroup of the identity component G of G . Let d be the dimension of the abelian variety G /L , and let m be the maximal integersuch that ℓ m | [ G : G ]. Choose an embedding of L into some GL n . Then every elementary abeliansubgroup of G has rank ≤ n + 2 d + m .To prove the theorem, we need a lemma on tangent spaces. Let S be an algebraic space, andlet X be an S -functor, that is, a presheaf of sets on AlgSp /S . Recall [49, II 3.1] that the tangentbundle to X is defined to be the S -functor T X/S = H om S (Spec( O S [ ǫ ] / ( ǫ )) , X ) , which is endowed with a projection to X . For every point u ∈ X ( S ), the tangent space to X at u is the S -functor [49, II 3.2] T uX/S = T X/S × X,u S. Recall [49, II 3.11] that, for S -functors Y and Z , we have an isomorphism T H om S ( Y,Z ) /S ≃ H om S ( Y, T
Z/S ) . For a morphism f : Y → Z of S -functors, this induces an isomorphism(5.3.1) T f H om S ( Y,Z ) /S ≃ H om Z/S (( Y, f ) , T Z/S ) . Z is an S -group, that is, a presheaf of groups on AlgSp /S . Then we have an iso-morphism of schemes T Z/S ≃ Z × S Lie(
Z/S ), where Lie(
Z/S ) = T Z/S . Thus (5.3.1) induces anisomorphism(5.3.2) T f H om S ( Y,Z ) /S ∼ −→ H om S ( Y, Lie(
Z/S )) . Furthermore, if Y is an S -group and f is a homomorphism of S -groups, then the image of T f H om S -group ( Y,Z ) /S by (5.3.2) is Z S ( Y, Lie(
Z/S )) [49, II 4.2], where Y acts on Lie( Z/S ) by theformula y Ad( f ( y )). Lemma 5.4.
Let f : Y → Z be a homomorphism of S -groups as above. Let c : Z → H om S - group ( Y, Z ) be the morphism given by z ( y ( z − f ( y ) z )) . Then the composition
Lie(
Z/S ) T c/S −−−→ T f H om S - group ( Y,Z ) /S → H om S ( Y, Lie(
Z/S )) is given by t ( y Ad( f ( y )) t − t ) , and the image is B S ( Y, Lie(
Z/S )) .Proof. The exact sequence 1 → Lie(
Z/S ) → T Z/S → Z → T Z/S with the semidirect product Lie(
Z/S ) ⋊ Z . An element ( t, z ) of the semidirect product (evaluated at an S -scheme S ′ ) corresponds tothe image of dR z ( t ) ∈ T zZ/S ( S ′ ), where R z : Z × S S ′ → Z × S S ′ is the right translation by z .Multiplication in the semidirect product is given by( t, z )( t ′ , z ′ ) = ( t + Ad( z ) t ′ , zz ′ ) . The image of t under T c/S in T f H om S ( Y,Z ) /S ∼ −→ H om Z/S ( Y, T
Z/S ) is T c/S ( t, y ( t, − (0 , f ( y ))( t,
1) = (Ad( f ( y )) t − t, f ( y )) . Hence the image in H om S ( Y, Lie(
Z/S )) is y Ad( f ( y )) t − t . Proof of Theorem 5.1.
We may assume k algebraically closed and G smooth. As in the beginningof Section 5, let u : A → G be a rational point of X , let H be the inertia at u , let Y = H \ G , and let c : G → X be the G -equivariant morphism sending g to ug , which factorizes through an immersion j : Y → X . Since H ( A, Lie( G )) = 0 for any action of A on Lie( G ), it follows from Lemma 5.4that T c : Lie( G ) → T uX is an epimorphism. Thus the map T { H } j : T { H } Y → T uX is an isomorphism.Since Y is smooth [49, VI B j is étale [22, Corollaire 17.11.2] and hence an open immersion atthis point. In other words, the orbit of u contains an open neighborhood of u . Since the rationalpoints of X form a dense subset [22, Corollaire 10.4.8], the orbits of rational points form an opencovering of X , which implies that X is smooth. Throughout this section κ is a field and k is an algebraically closed field. Definition 6.1.
For a functor F : C → D and an object d of D , let ( d ↓ F ) = C × D D d/ (strictfiber product) be the category whose objects are pairs ( c, φ ) of an object c of C and a morphism φ : d → F ( c ) in D , and arrows are defined in the natural way. Recall that F is said to be cofinal if, for every object d of D , the category ( d ↓ F ) is nonempty and connected.If F is cofinal and G : D → E is a functor such that lim −→ GF exists, then lim −→ G exists and themorphism lim −→ GF → lim −→ G is an isomorphism [33, Theorem IX.3.1]. For compatibility with [49, II 4.1], we write the adjoint action as left action. emma 6.2. Let F : C → D be a full and essentially surjective functor. Then F is cofinal.Proof. Let d be an object of D . As F is essentially surjective, there exist an object c of C and anisomorphism f : d ∼ −→ F ( c ) in D , which give an object of ( d ↓ F ). As F is full, for any morphism g : d → F ( c ′ ), with c ′ an object of C , there exists a morphism h : c → c ′ in C such that F ( h ) = gf − ,which gives a morphism ( c, f ) → ( c ′ , g ) in ( d ↓ F ).We now introduce some enriched categories, which will be of use in the structure theorems,especially Theorem 6.17. Definition 6.3.
Let D be a category enriched in the category AlgSp /κ of algebraic κ -spaces,with Cartesian product as the monoidal operation [29, Section 1.2]. For objects X and Y of D ,Hom D ( X, Y ) is an algebraic κ -space and composition of morphisms in D is given by morphisms ofalgebraic κ -spaces. We denote by D ( κ ) the category having the same objects as D , in whichHom D ( κ ) ( X, Y ) = (Hom D ( X, Y ))( κ ) . Assume that κ is separably closed. We denote by D ( π ) the category having the same objects as D , in which Hom D ( π ) ( X, Y ) = π (Hom D ( X, Y )) . Note that, if Hom D ( X, Y ) is of finite type, Hom D ( π ) ( X, Y ) is a finite set. We have a functor η : D ( κ ) → D ( π ) , which is the identity on objects, and sends f ∈ Hom D ( X, Y )( κ ) to the connected componentcontaining it. Assume that Hom D ( X, Y ) is locally of finite type. If κ is algebraically closed, or iffor all X , Y in D , Hom D ( X, Y ) is smooth over κ , then η is full, hence cofinal by Lemma 6.2. Construction 6.4.
Let G be an algebraic group over k , let X be an algebraic space of finitepresentation over k , endowed with an action of G , and let ℓ be a prime number. We define acategory enriched in the category Sch ft /k of schemes of finite type over k , A G,X,ℓ , as follows. Objects of A G,X,ℓ are pairs (
A, C ) where A is an elementary abelian ℓ -subgroup of G and C is a connected component of the algebraic space of fixed points X A (which is a closedalgebraic subspace of X if X is separated). For objects ( A, C ) and ( A ′ , C ′ ) of A G,X , we denoteby Trans G (( A, C ) , ( A ′ , C ′ )) the transporter of ( A, C ) into ( A ′ , C ′ ), namely the closed subgroupscheme of G representing the functor S
7→ { g ∈ G ( S ) | g − A S g ⊂ A ′ S , C S g ⊃ C ′ S } . In fact, Trans G (( A, C ) , ( A ′ , C ′ )) is a closed and open subscheme of the scheme Trans G ( A, A ′ )defined by the cartesian square Trans G ( A, A ′ ) / / (cid:15) (cid:15) Q a ∈ A A ′ (cid:15) (cid:15) G / / Q a ∈ A G where the lower horizontal arrow is given by g ( g − ag ) a ∈ A . Indeed, if we consider the morphism F : Trans G ( A, A ′ ) × X A ′ → X A ( g, x ) xg − and the induced map φ : π (Trans G ( A, A ′ )) → π ( X A ) Γ π ( F )(Γ , C ′ ) , G (( A, C ) , ( A ′ , C ′ )) is the union of the connected components of Trans G ( A, A ′ ) corre-sponding to φ − ( C ). We defineHom A G,X,ℓ (( A, C ) , ( A ′ , C ′ )) := Trans G (( A, C ) , ( A ′ , C ′ )) . Composition of morphisms is given by the composition of transportersTrans G (( A ′ , C ′ ) , ( A ′′ , C ′′ )) × Trans G (( A, C ) , ( A ′ , C ′ )) → Trans G (( A, C ) , ( A ′′ , C ′′ )) , which is a morphism of k -schemes. When no confusion arises, we omit ℓ from the notation. Wewill denote A G, Spec( k ) by A G .For an object ( A, C ) of A G,X , we denote by Cent G ( A, C ) its centralizer , namely the closedsubscheme of G representing the functor S
7→ { g ∈ G ( S ) | C S g = C S and g − ag = a for all a ∈ A } . For objects (
A, C ), ( A ′ , C ′ ) of A G,X , we have natural injections (cf. [37, (8.2)])(6.4.1) Cent G ( A, C ) \ Trans G (( A, C ) , ( A ′ , C ′ )) → Cent G ( A ) \ Trans G ( A, A ′ ) → Hom(
A, A ′ ) . We let A ♭G,X denote the category having the same objects as A G,X , but with morphisms definedby the left hand side of (6.4.1). We call the finite group(6.4.2) W G ( A, C ) := Cent G ( A, C ) \ Trans G (( A, C ) , ( A, C ))the
Weyl group of (
A, C ). This is a subgroup of the finite group W G ( A ) = Cent G ( A ) \ Norm G ( A ) ⊂ Aut( A ) . The functors A G,X ( k ) → A G,X ( π ) → A ♭G,X (the second one defined via (6.4.1)) are cofinal by Lemma 6.2.Let k ′ be an algebraically closed extension of k . We have a functor A G,X ( k ) → A G k ′ ,X k ′ ( k ′ )carrying ( A, C ) to (
A, C k ′ ). Since the map π (Trans G (( A, C ) , ( A ′ , C ′ )) → π (Trans G k ′ (( A, C k ′ ) , ( A ′ , C ′ k ′ )))is a bijection, this induces a functor A G,X ( π ) → A G k ′ ,X k ′ ( π ).In the rest of the section we assume ℓ invertible in k . Lemma 6.5.
The category A G,X ( π ) is essentially finite, and the functor A G,X ( π ) → A G k ′ ,X k ′ ( π ) is an equivalence. In particular, A G ( π ) is essentially finite.Proof. Let S be a set of representatives of isomorphisms classes of objects of A G ( π ). In otherwords, S is a set of representatives of conjugacy classes of elementary abelian ℓ -subgroups of G .By Corollary 5.3, this is a finite set. Let T be the set of objects ( A, C ) of A G,X ( π ) such that A ∈ S . Then T is a finite set. The conclusion follows from the following facts:(a) For ( A, C ) and ( A ′ , C ′ ) in A G,X , Hom A G,X ( π ) (( A, C ) , ( A ′ , C ′ )) is finite (Definition 6.3), and,by Construction 6.4,Hom A G,X ( π ) (( A, C ) , ( A ′ , C ′ )) ∼ −→ Hom A Gk ′ ,Xk ′ ( π ) (( A, C k ′ ) , ( A ′ , C ′ k ′ )) . (b) The finite set T is a set of representatives of isomorphism classes of objects of A G,X ( π ),and { ( A, C k ′ ) | ( A, C ) ∈ T } is a set of representatives of isomorphism classes of objects of A G k ′ ,X k ′ ( π ).Indeed, (b) follows from the following obvious lemma. Lemma 6.6.
Let B , C be sets endowed with equivalence relations denoted by ≃ and let f : B → C be a map such that b ≃ b ′ implies f ( b ) ≃ f ( b ′ ) . Let S be a set of representatives of C . For every s ∈ S , let T s be a set of representatives of f − ( s ) . Then S s ∈ S T s is a set of representatives of B ifand only if for every b ∈ B and every c ∈ S such that f ( b ) ≃ c , there exists b ′ ∈ f − ( c ) such that b ≃ b ′ . emark 6.7. Let G be an algebraic group over k and let T be a subtorus of G . Then W G ( T ) =Cent G ( T ) \ Norm G ( T ) is a finite subgroup of Aut( T ). The inclusionsNorm G ( T ) ⊂ Norm G ( T [ ℓ ]) , Cent G ( T ) ⊂ Cent G ( T [ ℓ ])induce a homomorphism ρ : W G ( T ) → W G ( T [ ℓ ]). Via the isomorphisms Aut( T ) ≃ Aut( M ) andAut( T [ ℓ ]) ≃ Aut(
M/ℓM ), where M = X ∗ ( T ), ρ is compatible with the reduction homomorphismAut( M ) → Aut(
M/ℓM ). If T is a maximal torus, then ρ is surjective by the proof of [43, 1.1.1].For ℓ > ρ is injective. In fact, for an element g of Ker(Aut( M ) → Aut(
M/ℓM )) and arbitrary ℓ , the ℓ -adic logarithm log( g ) := P ∞ m =1 ( − m − m ( g − m ∈ ℓ End( M ) ⊗ Z ℓ is well defined. If g n = idfor some n ≥
1, then n log( g ) = log( g n ) = 0, so that log( g ) = 0. In the case ℓ >
2, we then have g = exp log( g ) = id. For ℓ = 2, ρ is not injective in general. For example, if G = SL and T is amaximal torus, then W G ( T ) ≃ Z / W G ( T [2]) = { } .If G = GL n and T is a maximal torus, then ρ is an isomorphism for arbitrary ℓ . In fact, in thiscase, Norm G ( T ) = Norm G ( T [ ℓ ]) and Cent G ( T ) = Cent G ( T [ ℓ ]). Notation 6.8.
We will sometimes omit the constant coefficient F ℓ from the notation. We willsometimes write H ∗ G for H ∗ ( BG ) = H ∗ ( BG, F ℓ ). Construction 6.9.
Let T = Trans G ( A, A ′ ), let g ∈ T ( k ), and let c g : A → A ′ be the map a g − ag . In the above notation, the morphism Bc g : BA → BA ′ induces a homomorphism θ g : H ∗ A ′ → H ∗ A . This defines a presheaf ( H ∗ A , θ g ) on A ♭G , hence on A ♭G,X .If ( A, C ) is an object of A G,X , we have H ∗ ([ C/A ]) = H ∗ A ⊗ H ∗ ( C ) . The restriction H ∗ ([ X/G ]) → H ∗ ([ C/A ]) induced by the inclusion (
C, A ) → ( X, G ), composedwith the projection(6.9.1) H ∗ ([ C/A ]) → H ∗ A induced by H ∗ ( C ) → H ( C ) = F ℓ , defines a homomorphism(6.9.2) ( A, C ) ∗ : H ∗ ([ X/G ]) → H ∗ A . For g ∈ Trans((
A, C ) , ( A ′ , C ′ ))( k ) ⊂ T ( k ), we have the following 2-commutative square of grou-poids in the category AlgSp /U (Construction 1.1):( C ′ , A ) • (id ,c g ) / / ( g − , id) (cid:15) (cid:15) ☞☞☞☞ (cid:2) (cid:10) ( C ′ , A ′ ) • (cid:15) (cid:15) ( C, A ) • / / ( X, G ) • (with trivial action of A and A ′ on C ′ and trivial action of A on C ), where the 2-morphism is givenby g . The corresponding 2-commutative square of Artin stacks BA × C ′ / / (cid:15) (cid:15) BA ′ × C ′ (cid:15) (cid:15) BA × C / / [ X/G ]induces by adjunction (Notation 2.3) the following commutative square: H ∗ ([ X/G ]) / / (cid:15) (cid:15) H ∗ ([ C/A ]) [ g − / id] ∗ (cid:15) (cid:15) H ∗ ([ C ′ /A ′ ]) [id /c g ] ∗ / / H ∗ ([ C ′ /A ]) . H ∗ ([ X/G ]) ( A ′ ,C ′ ) ∗ (cid:15) (cid:15) ( A,C ) ∗ % % ❑❑❑❑❑❑❑❑❑❑ H ∗ A ′ θ / / H ∗ A . Therefore the maps (
A, C ) ∗ (6.9.2) define a homomorphism(6.9.3) a ( G, X ) : H ∗ ([ X/G ]) → lim ←− A ♭G,X ( H ∗ A , θ g ) . Note that the right-hand side is the equalizer of( j , j ) : Y ( A,C ) ∈A G,X H ∗ A ⇒ Y g : ( A,C ) → ( A ′ ,C ′ ) H ∗ A , where g runs through morphisms in A ♭G,X , j ( h ( A,C ) ) = ( h ( A,C ) ) g , j ( h ( A,C ) ) = ( θ g h ( A ′ ,C ′ ) ) g .Moreover, by the finiteness results Corollary 4.8 and Lemma 6.5, the right-hand side of (6.9.3) isa finite H ∗ ( BG )-module, and, in particular, a finitely generated F ℓ -algebra.To state our main result for the map a ( G, X ) (6.9.3), we need to recall the notion of uniform F -isomorphism. For future reference, we give a slightly extended definition as follows. Definition 6.10.
Let GrVec be the category of graded F ℓ -vector spaces. It is an F ℓ -linear ⊗ -category. The commutativity constraint of GrVec follows Koszul’s rule of signs, such that a (pseudo-)ring in GrVec is an anti-commutative graded F ℓ -(pseudo-)algebra.Let C be a category. As a special case of Construction 3.7, the functor category GrVec C :=Fun( C op , GrVec) is a F ℓ -linear ⊗ -category. The functor lim ←− C : GrVec C → GrVec is the right adjointto the unital ⊗ -functor GrVec → GrVec C , thus has a right unital ⊗ -structure. If u : R → S is ahomomorphism of pseudo-rings in GrVec C , we say that u is a uniform F -injection (resp. uniform F -surjection ) if there exists an integer n ≥ i of C and any homogeneouselement (or, equivalently, any element) a in the kernel of u i (resp. in S i ), a ℓ n = 0 (resp. a ℓ n is in theimage of u i ). Note that a ℓ n = 0 for some n ≥ a m = 0 for some m ≥
1. We say u is a uniform F -isomorphism if it is both a uniform F -injection and a uniform F -surjection. Thesedefinitions apply in particular to GrVec by taking C to be a discrete category of one object, inwhich case the notion of a uniform F -isomorphism coincides with the definition in [36, Section 3].The following result is an analogue of Quillen’s theorem ([36, Theorem 6.2], [37, Theorem 8.10]): Theorem 6.11.
Let X be a separated algebraic space of finite type over k , and let G be an algebraicgroup over k acting on X . Then the homomorphism a ( G, X ) (6.9.3) is a uniform F -isomorphism(Definition 6.10). Remark 6.12.
Let A be an elementary abelian ℓ -group of rank r ≥
0. We identify H ( BA, F ℓ )with ˇ A = Hom( A, F ℓ ). Recall [36, Section 4] that we have a natural identification of F ℓ -gradedalgebras H ∗ ( BA, F ℓ ) = ( S( ˇ A ) if ℓ = 2 ∧ ( ˇ A ) ⊗ S( β ˇ A ) if ℓ > ∧ ) denotes a symmetric (resp. exterior) algebra over F ℓ , and β : ˇ A → H ( BA, F ℓ )is the Bockstein operator. In particular, if { x , . . . , x r } is a basis of ˇ A over F ℓ , then H ∗ ( BA, F ℓ ) = ( F ℓ [ x , . . . , x r ] if ℓ = 2 ∧ ( x , . . . , x r ) ⊗ F ℓ [ y , . . . , y r ] if ℓ > y i = βx i . 34 orollary 6.13. With X and G as in Theorem 6.11, let K ∈ D bc ([ X/G ] , F ℓ ) . The Poincaré series PS t ( H ∗ ([ X/G ] , K )) = X i ≥ dim F ℓ H i ([ X/G ] , K ) t i is a rational function of t of the form P ( t ) / Q ≤ i ≤ n (1 − t i ) , with P ( t ) ∈ Z [ t ] . The order of thepole of PS t ( H ∗ ([ X/G ])) at t = 1 is the maximum rank of an elementary abelian ℓ -subgroup A of G such that X A = ∅ .Proof. By Theorem 4.6, H ∗ ([ X/G ] , K ) is a finitely generated module over H ∗ ([ X/G ]), which is afinitely generated algebra over F ℓ . Therefore the Poincaré series PS t ( H ∗ ([ X/G ] , K )) is a rationalfunction of t , and the order of the pole at t = 1 of PS t ( H ∗ ([ X/G ])) is equal to the dimension of thecommutative ring H ∗ ([ X/G ]). To show that PS t ( H ∗ ([ X/G ] , K )) is of the form given in Corollary6.13, recall (Remark 4.9) that we have shown in the proof of Theorem 4.6 that H ∗ ([ X/G ] , K ) is aquotient H ∗ ( BH )-module of H ∗ ([ X/H ] , f ∗ K ) for a certain affine subgroup H of G , f denoting thecanonical morphism [ X/H ] → [ X/G ]. Embedding H into some GL n and applying Corollary 4.7, wededuce that H ∗ ([ X/G ] , K ) is a finite H ∗ ( B GL n )-module. Since H ∗ ( B GL n ) ≃ F ℓ [ c , . . . , c n ], where c i is of degree 2 i (Theorem 4.4), PS t ( M ∗ ) is of the form P ( t ) / Q ≤ i ≤ n (1 − t i ) with P ( t ) ∈ Z [ t ] forevery finite graded H ∗ ( B GL n )-module M ∗ . The last assertion of Corollary 6.13 is derived fromTheorem 6.11 as in [36, Theorem 7.7]. One can also see it in a more geometric way, observing thatthe reduced spectrum of H ε ∗ ([ X/G ]) (where ε = 1 if ℓ = 2 and 2 otherwise) is homeomorphic toan amalgamation of standard affine spaces A = Spec( H ε ∗ A ) red associated with the objects ( A, C )of A G,X (see Construction 11.1).
Example 6.14.
Let G be a connected reductive group over k with no ℓ -torsion, and let T be amaximal torus of G . Let ι : A ′ ֒ → A ♭G be the full subcategory spanned by T [ ℓ ]. The functor ι iscofinal. Indeed, for every object A of A ♭G , since A is toral, there exists a morphism c g : A → T [ ℓ ]in A ♭G . Moreover, for morphisms c g : A → T [ ℓ ], c g ′ : A → T [ ℓ ] in A ♭G , there exists an isomor-phism c h : T [ ℓ ] → T [ ℓ ] such that c h c g = c g ′ in A ♭G , by [43, 1.1.1] applied to the conjugation c g − g ′ : c g ( A ) → c g ′ ( A ). Let W = W G ( T ). The map a ( G, Spec( k )) can be identified with theinjective F -isomorphism H ∗ G ≃ ( H ∗ T ) W → ( H ∗ T [ ℓ ] ) W induced by restriction (where the isomorphism is (4.11.2)). In particular,lim ←− A ∈A ♭G ( H ε ∗ A ) red ≃ S( T [ ℓ ] ∨ ) W , where ε = 1 if ℓ = 2 and ε = 2 if ℓ >
2. Moreover, for ℓ > a ( G, Spec( k )) induces an isomorphism H ∗ G ≃ (( H ∗ T [ ℓ ] ) red ) W . Example 6.15.
Let X = X (Σ) be a toric variety over k with torus T , where Σ is a fan in N ⊗ R and N = X ∗ ( T ). We identify T [ ℓ ] with N ⊗ µ ℓ . The inertia I σ ⊂ T of the orbit O σ correspondingto a cone σ ∈ Σ is N σ ⊗ G m , where N σ is the sublattice of N generated by N ∩ σ , so that A σ = I σ [ ℓ ] ≃ N σ ⊗ µ ℓ . The latter can be identified with the image of N ∩ σ in N ⊗ F ℓ . This definesan object ( A σ , C σ ) of A T,X , where C σ is the connected component of X A σ containing O σ . Thefunctor Σ → A ♭T,X is cofinal. Thus we have a canonical isomorphismlim ←− A ∈A ♭T,X ( H ε ∗ A ) red ≃ lim ←− σ ∈ Σ ( H ε ∗ A σ ) red . Note that ( H ∗ A σ ) red can be canonically identified with S( M σ ) ⊗ F ℓ , where M σ = M/ ( M ∩ σ ⊥ ) andS( M σ ) is the algebra of integral polynomial functions on σ . In particular, we have a canonicalisomorphism(6.15.1) lim ←− A ∈A ♭T,X ( H ε ∗ A ) red ≃ PP ∗ (Σ) ⊗ F ℓ , where PP ∗ (Σ) = { f : Supp(Σ) → R | ( f | σ ) ∈ S( M σ ) for each σ ∈ Σ }
35s the algebra of piecewise polynomial functions on Σ. Recall that Payne established an isomor-phism from the integral equivariant Chow cohomology ring A ∗ T ( X ) of Edidin and Graham [14,2.6] onto PP ∗ (Σ) [35, Theorem 1]. Combining Theorem 6.11 and (6.15.1), we obtain a uniform F -isomorphism H ∗ ([ X/T ] , F ℓ ) → PP ∗ (Σ) ⊗ F ℓ . If X is smooth, this is an isomorphism, and PP ∗ (Σ) is isomorphic to the Stanley-Reisner ring of Σ[3, Section 4].In the rest of this section, we state an analogue of Theorem 6.11 with coefficients. Construction 6.16.
Let G be an algebraic group over k , X an algebraic k -space endowed withan action of G , and K ∈ D +cart ([ X/G ] , F ℓ ).If A , A ′ are elementary abelian ℓ -subgroups of G and g ∈ G ( k ) conjugates A into A ′ (i.e. g − Ag ⊂ A ′ ), A acts trivially on X A ′ via c g = A → A ′ (where c g is the conjugation s g − sg ),and we have an equivariant morphism (1 , c g ) : ( X A ′ , A ) → ( X, G ), where 1 denotes the inclusion X A ′ ⊂ X , inducing [1 /c g ] : [ X A ′ /A ] = BA × X A ′ → [ X/G ] . We thus have, for all q , a restriction map H q ([ X/G ] , K ) → H q ([ X A ′ /A ] , [1 /c g ] ∗ K ) . On the other hand, we have a natural projection π : [ X A ′ /A ] = BA × X A ′ → X A ′ , hence an edge homomorphism for the corresponding Leray spectral sequence H q ([ X A ′ /A ] , [1 /c g ] ∗ K ) → H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) . By composition we get a homomorphism(6.16.1) a q ( A, A ′ , g ) : H q ([ X/G ] , K ) → H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) . Since R ∗ π ∗ F ℓ = L q R q π ∗ F ℓ is a constant sheaf of value H ∗ ( BA, F ℓ ), R ∗ π ∗ [1 /c g ] ∗ K = L q R q π ∗ [1 /c g ] ∗ K is endowed with a H ∗ ( BA, F ℓ )-module structure by Constructions 3.4 and 3.7, which induces a H ∗ ( BG, F ℓ )-module structure via the ring homomorphism [1 /c g ] ∗ : H ∗ ( BG, F ℓ ) → H ∗ ( BA, F ℓ ).The map a ( A, A ′ , g ) = L q a q ( A, A ′ , g ) is H ∗ ( BG, F ℓ )-linear.If ( Z, Z ′ , h ) is a second triple consisting of elementary abelian ℓ -subgroups Z , Z ′ , and h ∈ G ( k )such that c h : Z → Z ′ , the datum of elements a and b of G ( k ) such that g = ahb and c a : A → Z , c b : Z ′ → A ′ , defines a commutative diagram(6.16.2) A c g / / c a (cid:15) (cid:15) A ′ Z c h / / Z ′ , c b O O hence a morphism [ b − /c a ] : [ X A ′ /A ] → [ X Z ′ /Z ], fitting into a 2-commutative diagram(6.16.3) [ X A ′ /A ] [ b − /c a ] (cid:15) (cid:15) [1 /c g ] z z ✉✉✉✉✉✉✉✉✉✉ π / / X A ′ b − (cid:15) (cid:15) [ X/G ] ✺✺✺✺ (cid:22) (cid:30) [ X Z ′ /Z ] [1 /c h ] o o π / / X Z ′ , where the 2-morphism of the triangle is induced by b . Consider the homomorphism(6.16.4)( a, b ) ∗ : H ( X Z ′ , R q π ∗ [1 /c h ] ∗ K ) → H ( X A ′ , ( b − ) ∗ R q π ∗ [1 /c h ] ∗ K ) → H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) , b − and the second map is base change map for the squarein (6.16.3). This fits into a commutative triangle(6.16.5) H q ([ X/G ] , K ) (cid:15) (cid:15) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ H ( X Z ′ , R q π ∗ [1 /c h ] ∗ K ) ( a,b ) ∗ / / H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) , where the vertical and oblique maps are given by (6.16.1). Denote by(6.16.6) A G ( k ) ♮ the following category. Objects of A G ( k ) ♮ are triples ( A, A ′ , g ) as above, morphisms ( A, A ′ , g ) → ( Z, Z ′ , h ) are pairs ( a, b ) ∈ G ( k ) × G ( k ) such that g = ahb and c a : A → Z , c b : Z ′ → A ′ . Viathe maps ( a, b ) ∗ (6.16.4), the groups H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) form a projective system indexed by A G ( k ) ♮ , and by the commutativity of (6.16.5) we get a homomorphism(6.16.7) a qG ( X, K ) : H q ([ X/G ] , K ) → R qG ( X, K ) , where(6.16.8) R qG ( X, K ) := lim ←− ( A,A ′ ,g ) ∈A G ( k ) ♮ H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) . Since L q ( a, b ) ∗ is H ∗ ( BG, F ℓ )-linear, R ∗ G ( X, K ) := L q R qG ( X, K ) is endowed with a structure of H ∗ ( BG, F ℓ )-module. The map(6.16.9) a G ( X, K ) = M q a qG ( X, K ) : H ∗ ([ X/G ] , K ) → R ∗ G ( X, K )induced by (6.16.7) is a homomorphism of H ∗ ( BG, F ℓ )-modules. If K is a (pseudo-)ring in D +cart ([ X/G ] , F ℓ ), R ∗ G ( X, K ) is a F ℓ -(pseudo-)algebra and a G ( X, K ) is a homomorphism of F ℓ -(pseudo-)algebras. Theorem 6.17.
Let G be an algebraic group over k , X a separated algebraic space of finite typeover k endowed with an action of G , and K ∈ D + c ([ X/G ] , F ℓ ) .(a) R qG ( X, K ) is a finite-dimensional F ℓ -vector space for all q ; if K ∈ D bc ([ X/G ] , F ℓ ) , R ∗ G ( X, K ) is a finite module over H ∗ ( BG, F ℓ ) .(b) If K is a pseudo-ring in D + c ([ X/G ] , F ℓ ) (Construction 3.8), the kernel of the homomorphism a G ( X, K ) (6.16.9) is a nilpotent ideal of H ∗ ([ X/G ] , K ) . If, moreover, K is commutative,then a G ( X, K ) is a uniform F -isomorphism (Definition 6.10). Remark 6.18.
The projective limit in (6.16.7) is the equalizer of the double arrow( j , j ) : Y A ∈A G Γ( X A , R q π A ∗ [1 /c ] ∗ K ) ⇒ Y ( A,A ′ ,g ) ∈A G ( k ) ♮ Γ( X A ′ , R q π ( A,A ′ ,g ) ∗ [1 /c g ] ∗ K ) , where π A = π ( A,A, , [1 /c ] : [ X A /A ] → [ X/G ], j is induced by (1 , g ) : ( A, A ′ , g ) → ( A, A,
1) and j is induced by ( g,
1) : (
A, A ′ , g ) → ( A ′ , A ′ , C = A G ( k )). Let C be a category.Define a category C ♮ as follows. The objects of C ♮ are the morphisms A → A ′ of C . A morphism in C ♮ from A → A ′ to Z → Z ′ is a pair of morphisms ( A → Z, Z ′ → A ′ ) in C such that the followingdiagram commutes: A / / (cid:15) (cid:15) A ′ Z / / Z ′ . O O F be a presheaf of sets on C ♮ . Then the sequenceΓ( b C ♮ , F ) → Y A ∈C F (id A ) ⇒ Y ( a : A → A ′ ) ∈C ♮ F ( a )is exact. Here the two projections are induced by (id A , a ) : a → id A and ( a, id A ′ ) : a → id A ′ ,respectively.Indeed, because the two compositions are equal, we have a map s : Γ( b C ♮ , F ) → K , where K isthe equalizer of the double arrow. It is straightforward to check that the map K → Q a ∈C ♮ F ( a )factors through Γ( b C ♮ , F ) to give the inverse of s .Note that this statement generalizes the calculation of ends R A ∈C F ( A, A ) [33, Section IX.5] of afunctor F from C op × C to the category of sets. More generally, for any category D and any functor F : C op × C → D , R A ∈C F ( A, A ) can be identified with the limit lim ←− a : A → A ′ F ( A, A ′ ) indexed by C ♮ . Remark 6.19.
For K = F ℓ , the commutative diagram H ∗ ([ X/G ] , F ℓ ) / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ Q ( A,C ) ∈A G,X H ∗ A / / / / ≃ (cid:15) (cid:15) Q g : ( A,C ) → ( A ′ ,C ′ ) H ∗ A ≃ (cid:15) (cid:15) Q A ∈A G Γ( X A , R q π A ∗ F ℓ ) / / / / Q ( A,A ′ ,g ) ∈A G ( k ) ♮ Γ( X A ′ , R q π ( A,A ′ ,g ) ∗ F ℓ )induces a commutative diagram H ∗ ([ X/G ] , F ℓ ) a ( G,X ) / / a G ( X, F ℓ ) ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ lim ←− A ♭G,X H ∗ A ≃ (cid:15) (cid:15) R ∗ G ( X, F ℓ ) . Therefore Theorem 6.17 generalizes Theorem 6.11.Part (b) of Theorem 6.17 will be proved as a corollary of a more general structure theorem(Theorem 8.3). Part (a) will follow from the next lemma.
Lemma 6.20.
Let E G be the category enriched in Sch ft /k having the same objects as A G ( k ) ♮ andin which Hom E G (( A, A ′ , g ) , ( Z, Z ′ , h )) is the subscheme of G × G representing the presheaf of setson AlgSp /k : S
7→ { ( a, b ) ∈ ( G × G )( S ) | a − A S a ⊂ Z S , b − Z ′ S b ⊂ A ′ S , g = ahb } (so that by definition E G ( k ) = A G ( k ) ♮ ).(a) The functor F : E G ( π ) → A G ( π ) ♮ carrying ( A, A ′ , g ) to ( A, A ′ , γ ) , where γ is the connectedcomponent of Trans G ( A, A ′ ) containing g , is an equivalence of categories. In particular, E G ( π ) is equivalent to a finite category, and for every algebraically closed extension k ′ of k ,the natural functor E G ( π ) → E G k ′ ( π ) is an equivalence of categories.(b) The projective system H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) indexed by ( A, A ′ , g ) ∈ A G ( k ) ♮ factors through E G ( π ) . Remark 6.21.
The projective system in Lemma 6.20 (b) does not factor through ( A ♭G ) ♮ in gen-eral. Indeed, if G is a finite discrete group of order prime to ℓ , then A G ( π ) and A G ( π ) ♮ areboth connected groupoids of fundamental group G , while A ♭G is a simply connected groupoid. If K ∈ Mod c ( BG, F ℓ ), then the projective system in Lemma 6.20 (b) can be identified with the F ℓ -representation of G corresponding to K .The proof of Lemma 6.20 will be given after Remark 6.26. We will exploit the fact that thefamily of stacks [ X A ′ /A ] parameterized by ( A, A ′ , g ) ∈ A G ( k ) ♮ underlies a family “algebraicallyparameterized” by E G . To make sense of this, the following general framework will be convenient.38 efinition 6.22. Let D be a category enriched in AlgSp /κ (Definition 6.3). By a family of Artin κ -stacks parameterized by D , or, for short, an Artin D -stack , we mean a collection X = ( X A , x A,B , σ A , γ A,B,C ) A,B,C ∈D , where X A is an Artin stack over κ , x A,B : X A × Hom D ( A, B ) → X B is a morphism of Artin stacksover κ , σ A and γ A,B,C are 2-morphisms: X A id XA × i / / id ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ X A × Hom D ( A, A ) x A,A (cid:15) (cid:15) X A ✎✎✎✎ C K σ A X A × Hom D ( A, B ) × Hom D ( B, C ) x A,B × id Hom D ( B,C ) (cid:15) (cid:15) id XA × c / / γ A,B,C X A × Hom D ( A, C ) x A,C (cid:15) (cid:15) X B × Hom D ( B, C ) x B,C / / X C , ✖✖✖✖ G O satisfying identities of 2-morphisms expressing the unit and associativity axioms. Here i : Spec( κ ) → Hom D ( A, A ) is the unit section and c : Hom D ( A, B ) × Hom D ( B, C ) → Hom D ( A, C ) is the compo-sition.A morphism f : X → Y of Artin D -stacks is a collection (( f A ) A ∈D , ( φ A,B ) A,B ∈D ), where f A : X A → Y A is a morphism of Artin stacks over κ and φ A,B is a 2-morphism: X A × Spec( κ ) Hom D ( A, B ) x A,B / / f A × id (cid:15) (cid:15) ✑✑✑✑ (cid:4) (cid:12) φ A,B X Bf B (cid:15) (cid:15) Y A × Spec( κ ) Hom D ( A, B ) y A,B / / Y B satisfying certain identities of 2-morphisms with respect to the unit section i and the composition c . Definition 6.23.
Let Λ be a commutative ring and let X be an Artin D -stack. We define acategory D cart ( X, Λ) as follows. An object of D cart ( X, Λ) is a collection (( K A ) A ∈D , ( α A,B ) A,B ∈D ),where K A ∈ D cart ( X A , Λ), α A,B : x ∗ A,B K B → p ∗ K A , p : X A × Hom D ( A, B ) → X A is the projection,such that the following diagrams commute i ∗ x ∗ A,A K Ai ∗ α A,A / / σ ∗ A ≃ % % ▲▲▲▲▲▲▲▲▲▲ i ∗ p ∗ K A ≃ (cid:15) (cid:15) x ∗ A,C K C α
A,C / / γ ∗ A,B,C ≃ (cid:15) (cid:15) p ∗ K A K A x ∗ A,B x ∗ B,C K C α
B,C / / p ∗ x ∗ A,B K B . α A,B O O A morphism K → L in D cart ( X, Λ) is a collection ( K A → L A ) A ∈D of morphisms in D cart ( X A , Λ)commuting with α A,B . If S is an Artin stack over κ , we denote by S D the constant Artin D -stack. If κ is separably closed and Hom D ( A, B ) is noetherian for every A and every B in D , thenMod cart (Spec( κ ) D , Λ) is equivalent to the category of projective systems of Λ-modules indexed by D ( π ). Indeed, in this case, α A,B : p ∗ K B → p ∗ K A is a morphism between constant sheaves onHom D ( A, B ), and has to be constant on every connected component of Hom D ( A, B ). Remark 6.24. If D is discrete (i.e. induced from a usual category) and X is a D -scheme, i.e.a functor from D to the category of κ -schemes, the category D cart ( X, Λ) consists of families ofobjects K A ∈ D ( X A , Λ) and compatible transition maps X ∗ f K B → K A for f : A → B , and shouldnot be confused with the derived category of sheaves of Λ-modules on the total étale topos of X over D . 39 onstruction 6.25. Let f = (( f A ) A ∈D , ( φ A,B ) A,B ∈D ) be a morphism of Artin D -stacks. Thefunctors f ∗ A induce a functor f ∗ : D cart ( Y, Λ) → D cart ( X, Λ). On the other hand, for K ∈ D cart ( X, Λ) we have a diagram(6.25.1) y ∗ A,B Rf B ∗ K B (cid:15) (cid:15) p ∗ Rf A ∗ K A (cid:15) (cid:15) R ( f A × id) ∗ x ∗ A,B K B α
A,B / / R ( f A × id) ∗ p ∗ K A where the left (resp. right) vertical arrow is base change for the square φ A,B (resp. for the obviouscartesian square).Assume that Λ is annihilated by an integer invertible in κ , and that the condition (a) (resp.(b)) below holds:(a) Hom D ( A, B ) is smooth over κ for all objects A , B in D ;(b) f A is quasi-compact and quasi-separated and K A ∈ D +cart for every object A of D .Then the right vertical arrow is an isomorphism by smooth base change (resp. generic basechange (Remark 2.12)) from Spec( κ ) to Hom D ( A, B ), and thus the diagram (6.25.1) defines amap y ∗ A,B Rf B ∗ K B → p ∗ Rf A ∗ K A . These maps endow ( Rf A ∗ K A ) with a structure of object of D cart ( Y, Λ). We thus get a functor Rf ∗ : D cart ( X, Λ) → D cart ( Y, Λ) (resp. D +cart ( X, Λ) → D +cart ( Y, Λ)) . The adjunctions id D cart ( X A , Λ) → Rf A ∗ f ∗ A induce a natural transformation id → Rf ∗ f ∗ . Remark 6.26.
The construction of Rf ∗ above encodes the homotopy-invariance of étale coho-mology [52, XV Lemme 2.1.3]. More precisely, assume κ separably closed. Let Y, Y ′ be two Artinstacks over κ , L ∈ D cart ( Y, Λ), L ′ ∈ D cart ( Y ′ , Λ). A morphism c : ( Y, L ) → ( Y ′ , L ′ ) is a pair ( g, φ ),where g : Y → Y ′ , φ : g ∗ L ′ → L . Following [52, XV Section 2.1], we say that two morphisms c , c : ( Y, L ) → ( Y ′ , L ′ ) are homotopic if there exists a connected scheme T of finite type over κ ,two points 0 , ∈ T ( κ ), a morphism ( Y × Spec( κ ) T, pr ∗ L ) → ( Y, L ′ ) inducing c and c by takingfibers at 0 and 1, respectively. This is equivalent to the existence of an Artin D T -stack X and anobject K ∈ D cart ( X, Λ) such that X A = Y , X A ′ = Y ′ , K A = L , K A ′ = L ′ and inducing c and c by taking fibers at 0 and 1. Here D T is the Sch ft /κ -enriched category with Ob( D T ) = { A, A ′ } ,Hom D T ( A, A ) = Hom D T ( A ′ , A ′ ) = Spec( κ ), Hom D T ( A ′ , A ) = ∅ and Hom D T ( A, A ′ ) = T . If c and c are homotopic, then c ∗ = c ∗ : H ∗ ( Y ′ , L ′ ) → H ∗ ( Y, L ). To prove this, we may assume that T isa smooth curve as in [52, XV Lemme 2.1.3]. Let a : X → Spec( κ ) D T be the projection. By theabove, R ∗ a ∗ K is a projective system of graded Λ-modules indexed by D T ( π ), and c ∗ = c ∗ is theimage of the nontrivial arrow of D T ( π ). Proof of Lemma 6.20.
By construction, F is essentially surjective. Consider the morphism ofschemes φ : Hom E G (( A, A ′ , g ) , ( Z, Z ′ , h )) → Hom A G ( Z ′ , A ′ ) = Trans G ( Z ′ , A ′ ) given by ( a, b ) b .It fits into the following Cartesian diagramHom E G (( A, A ′ , g ) , ( Z, Z ′ , h )) (cid:15) (cid:15) φ / / Trans G ( Z ′ , A ′ ) (cid:15) (cid:15) { t ∈ Hom( Z ′ , A ′ ) | t ( c h ( Z )) ⊃ c g ( A ) } (cid:31) (cid:127) / / Hom( Z ′ , A ′ ) . In particular, φ is an open and closed immersion and induces an injection onHom E G ( π ) (( A, A ′ , g ) , ( Z, Z ′ , h )) → Hom A G ( π ) ( Z ′ , A ′ ) . In other words, the composite functor p ◦ F : E G ( π ) → A G ( π ) op is faithful, where p : A G ( π ) ♮ →A G ( π ) op . Therefore, F is faithful. To show that F is full, let ( α, β ) : F ( A, A ′ , g ) → F ( Z, Z ′ , h ) bea morphism in A G ( π ) ♮ . Choose b ∈ β ( k ) ⊂ G ( k ). Then we have a Cartesian diagramTrans G ( A, Z ) ψ / / (cid:15) (cid:15) Trans G ( A, A ′ ) (cid:15) (cid:15) Hom(
A, Z ) (cid:31) (cid:127) / / Hom(
A, A ′ ) , ψ : a ahb . In particular, ψ is an open and closed immersion. The map π (Trans G ( A, Z )) → π (Trans G ( A, A ′ )) induced by ψ carries α to γ = αηβ , where γ ∈ π (Trans G ( A, A ′ )) and η ∈ π (Trans G ( Z, Z ′ )) are the connected components of g and h , respectively. Thus there exists a ∈ α ( k ) ⊂ G ( k ) such that g = ψ ( a ) = ahb . Then ( a, b ) : ( A, A ′ , g ) → ( Z, Z ′ , h ) is a morphism in E G ( k ) = A G ( k ) ♮ , and induces a morphism τ in E G ( π ) such that F ( τ ) = ( α, β ). Therefore, F isan equivalence of categories. The second assertion of (a) follows from this and Lemma 6.5.Let us prove (b). For ( A, A ′ , g ) and ( Z, Z ′ , h ) in E G ( k ), consider the scheme T = Hom E G (( A, A ′ , g ) , ( Z, Z ′ , h ))and the tautological section t = ( a, b ) ∈ T ( T ). Then, if [ X A ′ /A ] T (resp. [ X Z ′ /Z ] T ) denotesthe product of [ X A ′ /A ] (resp. [ X Z ′ /Z ]) with T over Spec( k ), t defines a morphism of stacks[ b − /c a ] : [ X A ′ /A ] T → [ X Z ′ /Z ] T over T , whose fiber at ( a, b ) is [ b − /c a ]. These morphisms arecompatible with composition of morphisms up to 2-morphisms, and define a structure of E G -stack(Definition 6.22) on the family of stacks [ X A ′ /A ] for ( A, A ′ , g ) ∈ E G ( k ). Moreover, we have adiagram over T (6.26.1) [ X A ′ /A ] T [ b − /c a ] (cid:15) (cid:15) [1 /c g ] y y ssssssssss π / / X A ′ Tb − (cid:15) (cid:15) [ X/G ] T [ X Z ′ /Z ] T [1 /c h ] o o π / / X Z ′ T , where the 2-morphism of the triangle is induced by b . The fiber of (6.26.1) at ( a, b ) is (6.16.3).Therefore we get morphisms of Artin E G -stacks[ X/G ] E G ← ([ X A ′ /A ]) ( A,A ′ ,g ) π −→ ( X A ′ ) ( A,A ′ ,g ) . Thus the system H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) indexed by ( A, A ′ , g ) ∈ A G ( k ) ♮ can be extended to anobject of Mod cart (Spec( k ) E G , F ℓ ), which amounts to a system indexed by E G ( π ). More concretely,the morphism ( a, b ) ∗ (6.16.4) is the stalk at ( a, b ) of a morphism of constant sheaves on T (6.26.2) ( a, b ) ∗ : H ( X Z ′ , R q π ∗ [1 /c h ] ∗ K ) T → H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) T , defined by ( a, b ) via (6.26.1). Therefore it depends only on the connected component of ( a, b ) in T . We need the following lemma for the proof of Theorem 6.17 (a). Lemma 6.27.
Let Y be an algebraic space over k , and let A be a finite discrete group. Let L ∈ D bc ([ Y /A ] , F ℓ ) , where A acts trivially on Y . Let π : [ Y /A ] = BA × Y → Y be the secondprojection. Consider the structure of H ∗ ( BA, F ℓ ) -module on R ∗ π ∗ L given by Constructions 3.4and 3.7, as R ∗ π ∗ F ℓ is a constant sheaf of value H ∗ ( BA, F ℓ ) . Then R ∗ π ∗ L is a sheaf of constructible H ∗ ( BA, F ℓ ) -modules.Proof. We may assume L concentrated in degree zero. Suppose first that L is locally constant.Then R ∗ π ∗ L is a locally constant, constructible sheaf of H ∗ ( BA, F ℓ )-modules. Indeed, by definitionthere is an étale covering ( U α ) of Y such that L | [ U α /A ] (considered as a sheaf of F ℓ [ A ]-moduleson U α ) is a constant F ℓ [ A ]-module of finite dimension over F ℓ of value L α . Then R ∗ π ∗ L | U α isa constant H ∗ ( BA, F ℓ )-module of value H ∗ ( BA, L α ). By Theorem 4.6, H ∗ ( BA, L α ) is a finite H ∗ ( BA, F ℓ )-module, so the lemma is proved in this case. In general, we may assume Y to bean affine scheme. Take a finite stratification Y = S Y α into disjoint locally closed constructiblesubsets such that L | Y α is locally constant, or equivalently, that L | [ Y α /A ] is locally constant.Then, if π α = π | [ Y α /A ] → Y α , ( R ∗ π ∗ L ) | Y α ≃ Rπ α ∗ ( L | Y α ) by the finiteness of A , and we concludeby the preceding case. Proof of Theorem 6.17 (a).
By Lemma 6.20 (b) we can rewrite R qG ( X, K ) in the form R qG ( X, K ) := lim ←− ( A,A ′ ,g ) ∈E G ( π ) H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) . E G ( π ) is essentially finite (Lemma 6.20 (a)) and R q π ∗ [1 /c g ] ∗ K is constructible, the first asser-tion follows. Let us now prove the second assertion. As E G ( π ) is equivalent to a finite category, it isenough to show that, for all ( A, A ′ , g ), H ( X A ′ , R ∗ π ∗ [1 /c g ] ∗ K ) is a finite H ∗ ( BG, F ℓ )-module. As A acts trivially on X A ′ , R ∗ π ∗ [1 /c g ] ∗ K is a constructible sheaf of H ∗ ( BA, F ℓ )-modules by Lemma6.27. Therefore H ( X A ′ , R ∗ π ∗ [1 /c g ] ∗ K ) is a finite H ∗ ( BA, F ℓ )-module, thus, by Corollary 4.8, afinite H ∗ ( BG, F ℓ )-module. In this section we discuss two kinds of points of Artin stacks which will be of use to us:(a) geometric points , which generalize the usual geometric points of schemes,(b) ℓ -elementary points , which depend on a prime number ℓ , and are adapted to the study of themaps a ( G, X ) (6.9.3) and a G ( X, K ) (Theorem 6.17).The statement of the main structure theorem on Artin stacks (Theorem 8.3) requires only thenotion (b). The notion (a) is a technical tool used in the proof.
Definition 7.1.
Let X be a Deligne-Mumford stack. By a geometric point of X we mean amorphism x → X , where x is the spectrum of a separably closed field. The geometric points of X form a category P X , where a morphism from x → X to y → X is defined as an X -morphism X ( x ) → X ( y ) of thecorresponding strict henselizations [31, Remarque 6.2.1]. The category P X is essentially U -small.One shows as in [50, VIII Théorème 7.9] that the functor ( x → X ) ( F 7→ F x ) from P X to thecategory of points of the étale topos X et is an equivalence of categories.When X is a scheme, P X is the usual category of geometric points of X . If X = Spec k , k afield, P X is a connected groupoid whose fundamental group is isomorphic to the Galois group of k .As P X is an essentially U -small category, we have a morphism of topoi(7.1.1) p : c P X → X et , where c P X denotes the topos of presheaves on P X . For a sheaf F on X , p ∗ F is the presheaf( x → X )
7→ F x on P X , and p ∗ applied to a presheaf ( K x ) x ∈ P X is the sheaf whose set of sectionson U is lim ←− x ∈ P U K x . In particular we have an adjunction map(7.1.2) b X , F : F → p ∗ p ∗ F , which is a monomorphism, as X et has enough points and p ∗ b X , F is a split monomorphism (thisfact holds of course more generally for any topos X with an essentially small conservative familyof points P X , cf. [50, IV 6.7]). Proposition 7.2.
Let X be a locally noetherian Deligne-Mumford stack, Λ a noetherian commuta-tive ring, F a constructible sheaf of Λ -modules on X . Then the adjunction map b X , F : F → p ∗ p ∗ F (7.1.2) is an isomorphism. In particular, the homomorphism (7.2.1) φ : F ( X ) → lim ←− x ∈ P X F x is an isomorphism.Proof. If f : Y → X is a morphism of Deligne-Mumford stacks, the square of topoi c P Y p Y / / P f (cid:15) (cid:15) Y et f et (cid:15) (cid:15) c P X p X / / X et p ∗X f ∗ → P f ∗ p ∗Y and(7.2.3) f ∗ p X ∗ → p Y∗ P ∗ f and commutative diagrams(7.2.4) f ∗ G f ∗ b Y , G / / b X ,f ∗G (cid:15) (cid:15) f ∗ p Y∗ p ∗Y G ≃ (cid:15) (cid:15) f ∗ F b Y ,f ∗F / / f ∗ b X , F (cid:15) (cid:15) p Y∗ p ∗Y f ∗ F ≃ (cid:15) (cid:15) p X ∗ p ∗X f ∗ G (7.2.2) / / p X ∗ P f ∗ p ∗Y G f ∗ p X ∗ p ∗X F (7.2.3) / / p Y∗ P ∗ f p ∗X F . If f is a closed immersion, (7.2.2) is an isomorphism. If f is étale, (7.2.3) is an isomorphism.Let i : Z → X be a closed immersion, and let j : U → X be the complementary open immersion.Then the following diagram with exact rows commutes (where we write p for p X ):0 / / j ! j ∗ F / / b X ,j ! j ∗F (cid:15) (cid:15) F / / b X , F (cid:15) (cid:15) i ∗ i ∗ F / / b X ,i ∗ i ∗F (cid:15) (cid:15) / / p ∗ p ∗ j ! j ∗ F / / p ∗ p ∗ F / / p ∗ p ∗ i ∗ i ∗ F . Thus, to show that b X , F is an isomorphism, it suffices to show that both b X ,i ∗ i ∗ F and b X ,j ! j ∗ F areisomorphisms. By the square on the left of (7.2.4) applied to G = i ∗ F , b X ,i ∗ i ∗ F is an isomorphismif b Z ,i ∗ F is an isomorphism. On the other hand, the following diagram commutes:(7.2.5) j ∗ F b U ,j ∗F (cid:15) (cid:15) j ∗ j ! j ∗ F ∼ o o j ∗ b X ,j ! j ∗F / / b U ,j ∗ j ! j ∗F (cid:15) (cid:15) j ∗ p ∗ p ∗ j ! j ∗ F (7.2.3) ≃ (cid:15) (cid:15) p U∗ p ∗U j ∗ F p U∗ p ∗U j ∗ j ! j ∗ F ∼ o o ∼ / / p U∗ P ∗ j p ∗ j ! j ∗ F . We now prove that(7.2.6) i ∗ ( p ∗ p ∗ j ! j ∗ F ) = 0 . By the commutativity of (7.2.5), this will imply that b X ,j ! j ∗ F is an isomorphism if b U ,j ∗ F is anisomorphism. For any geometric point z → Z ,(7.2.7) ( p ∗ p ∗ j ! j ∗ F ) z ≃ lim −→ U ∈ N X ( z ) op lim ←− u ∈ P U ( j ! j ∗ F ) u , where N X ( z ) is the category of étale neighborhoods of z in X that are quasi-compact and quasi-separated schemes. Let U be any such neighborhood. Take a finite stratification ( U α ) α ∈ A of U by connected locally closed constructible subschemes such that the restrictions F | U α are locallyconstant. Let P U, ( U α ) α ∈ A be the category obtained from P U by inverting all arrows in the fullsubcategories P U α . Geometric points of the same stratum are isomorphic in P U, ( U α ) α ∈ A . Let B ⊂ A be the subset of indices α such that there exists a morphism from a geometric point of U α to z in P U, ( U α ) α ∈ A . Let V = S α ∈ B U α . Since the geometric points of V are closed undergenerization in U , V is an open subset of U . Since specialization maps on the same stratumare isomorphisms, the projective system (( j ! j ∗ F ) v ) v ∈ P V factors uniquely through a projectivesystem (( j ! j ∗ F ) x ) x ∈ P V, ( Uα ) α ∈ B (where on each stratum U α , α ∈ B all specialization maps areisomorphisms) and lim ←− v ∈ P V ( j ! j ∗ F ) v ≃ lim ←− x ∈ P V, ( Uα ) α ∈ B ( j ! j ∗ F ) x
43y Lemma 7.3 below. Note that P V contains z and that for any object x of P V, ( U α ) α ∈ B there existsa morphism from x to z . Therefore, as ( j ! j ∗ F ) z = 0, this limit is zero. This implies that thefull subcategory of N X ( z ) op consisting of the neighborhoods U such that lim ←− v ∈ P U ( j ! j ∗ F ) v = 0 iscofinal. It follows that the limit (7.2.7) is zero and hence (7.2.6) holds, as claimed. To sum up, wehave shown that b X , F is an isomorphism if both b Z ,i ∗ F and b U ,j ∗ F are isomorphisms.By induction, we may therefore assume F locally constant. Using the square on the right of(7.2.4), we may assume F constant. In this case it suffices to show that (7.2.1) is an isomorphism.We may further assume that X is connected and noetherian. Then P X is a connected categoryand the assertion is trivial. Lemma 7.3.
Let C be a category and let S be a set of morphisms in C . If we denote by F : C → S − C the localization functor, then F and F op are cofinal (Definition 6.1).Proof. It suffices to show that F is cofinal. Let X be an object of S − C , let Y be an object of C and let f : X → F Y be a morphism in S − C . Then f = t n s − n . . . t s − with t i in C and s i ∈ S .Using t i and s i , f can be connected to id X : X → F X in ( X ↓ S − C ). Remark 7.4.
If, in Proposition 7.2, the sheaf F is not assumed constructible, then the monomor-phism φ is not an isomorphism in general, as shown by the following example. Let X be a schemeof dimension ≥ k and let F = L x ∈| X | i x ∗ Λ, where | X | is the set of closed points of X and let i x : { x } → X be the inclusion. Then Γ( X, F ) ≃ Λ ( | X | ) (bycommutation of Γ( X, − ) with filtered inductive limits). On the other hand, for x ∈ P X , F x = Λ ifthe image of x is a closed point, and F x = 0 otherwise, hence lim ←− x ∈ P X Λ ≃ Λ | X | . The monomor-phism ϕ in Proposition 7.2 can be identified with the inclusion Λ ( | X | ) ⊂ Λ | X | , which is not anisomorphism, as | X | is infinite. Remark 7.5.
In the situation of Proposition 7.2, the morphism R Γ( X , F ) → R lim ←− x ∈ P X F x is not an isomorphism in general. In fact, if X = Spec( k ), then the left hand side computesthe continuous cohomology of the Galois group G of k while the right hand side computes thecohomology of G as a discrete group. Definition 7.6.
Let X be an Artin stack. By a geometric point of X we mean a morphism a : S → X , where S is a strictly local scheme. If a : S → X and b : S → X are geometric points of X , a morphism ( a : S → X ) → ( b : T → X ) is a morphism u : S → T together with a 2-morphism(7.6.1) S / / ❆❆❆❆❆❆❆ T (cid:15) (cid:15) X . ✁✁✁✁ < D We thus get a full subcategory P ′X of AlgSp / X (Notation 2.1). We define the category of geometricpoints of X as the category P X = M − X P ′X , localization of P ′X by the set M X of morphisms ( a → b ) in P ′X sending the closed point of S to theclosed point of T .Although P ′X is a U -category and not essentially small in general, we will show in Proposition7.9 that P X is essentially small. The next proposition shows that the definition above is consistentwith Definition 7.1. Proposition 7.7.
For any Deligne-Mumford stack X , the functor P X → P ′X sending every ge-ometric point x → X to the strict henselization X ( x ) → X induces an equivalence of categories ι : P X → P X . roof. Consider the functor F ′ : P ′X → P X sending S → X to its closed point s → X . For anymorphism in P ′X as in (7.6.1), its image under F ′ is the induced morphism X ( s ) → X ( t ) , where s and t are the closed points of S and T , respectively. The functor F : P X → P X induced by F ′ givesa quasi-inverse to ι : P X → P X . In fact F ι = id P X and we have a natural isomorphism id P X → ιF given by the morphism S → X ( s ) in M X for S → X in P X of closed point s . Remark 7.8.
The reason why we do not consider the category of points Point( X sm ) of the smoothtopos X sm is that already in the case X is an algebraic space, the functor Point( X sm ) → Point( X et )induced by the morphism of topoi ǫ : X sm → X et is not an equivalence. For example, if U → X isa smooth morphism and y is a geometric point of U lying above a geometric point x of X suchthat the image of y in the fiber U × X x is not a closed point, then the points ˜ x : F 7→ ( F X ) x and˜ y : F 7→ ( F U ) y of X sm are not equivalent, but have equivalent images in Point( X et ). Indeed, if wedenote by ǫ ! : X et → X sm the right adjoint of ǫ ∗ , then the stalk of ǫ ! G is G x at ˜ x , but is e at ˜ y . Proposition 7.9.
Let X be an Artin stack, and let ˜ P ′X be the full subcategory of P ′X consisting ofmorphisms S → X , such that S → X is the strict henselization of some smooth atlas X → X atsome geometric point of X . Let ˜ M X = M X ∩ Ar( ˜ P ′X ) . Then the inclusion ˜ P ′X ⊂ P ′X induces anequivalence of categories ˜ M − X ˜ P ′X → P X . Note that ˜ P ′X and hence ˜ P X are essentially small. Thus Proposition 7.9 shows that P X isessentially small, Proof.
We write ˜ P X = ˜ M − X ˜ P ′X . For x : S → X in P ′X , let A x be the full subcategory of(AlgSp / X ) x/ consisting of diagrams(7.9.1) S / / x (cid:31) (cid:31) ❅❅❅❅❅❅❅ X p (cid:15) (cid:15) X (cid:0)(cid:0)(cid:0)(cid:0) < D such that p is a smooth atlas. Then A x is nonempty since every smooth surjection to S admits asection [22, Corollaire 17.16.3 (ii)]. Moreover, A x admits finite nonempty products. Consider thefunctor F x : A x → ˜ P ′X sending (7.9.1) to the strict localization X ( s ) → X at the closed point s of S . For any pair of morphisms ( f, g ) : X ⇒ Y with the same source and target in A x , F x ( f ) and F x ( g ) have the same image in ˜ P X . Indeed, f | S = g | S implies F x ( f ) t = F x ( g ) t , where t ∈ ˜ M X is the inclusion of the closed point of X ( s ) . Thus there exists a unique functor G x making thefollowing diagram commutative A x F x / / (cid:15) (cid:15) ˜ P ′X (cid:15) (cid:15) | A x | G x / / ˜ P X where | A x | is the simply connected groupoid having the same objects as A x . This construction isfunctorial in x , in the sense that for x → y in P ′X , we have a natural transformation | A y | (cid:15) (cid:15) G y ! ! ❉❉❉❉❉❉❉❉ | A x | G x / / ˜ P X . ☎☎☎☎ > F Choosing an object X in A x for every x , we obtain a functor P ′X → ˜ P X sending x to X ( s ) . Thisfunctor factors through P X → ˜ P X and defines a quasi-inverse of ˜ P X → P X . Remark 7.10.
For any morphism f : X → Y of Artin stacks, composition with f defines a functor P ′ f : P ′X → P ′Y , which induces P f : P X → P Y . 45a) If f is a schematic universal homeomorphism, then P f is an equivalence of categories. Infact, for any object T → Y of P ′Y , the base change S = T × Y X → T is a schematic universalhomeomorphism, so that S is a strictly local scheme by [22, Proposition 18.8.18 (i)]. Thefunctor P ′Y → P ′X carrying T → Y to T × Y X → X carries M Y to M X and induces aquasi-inverse of P f .(b) For morphisms X → Y and
Z → Y of Artin stacks, the functor P ′X × Y Z → P ′X × P ′Y P ′Z is anequivalence of categories. Example 7.11.
Let k be a separably closed field, and let G be an algebraic group over k . Then P BG is a connected groupoid whose fundamental group is isomorphic to π ( G ).To prove this, by Remark 7.10 (a), we may assume k algebraically closed and G smooth.Then, for every object S → BG of P ′ BG , the corresponding G S -torsor is trivial and we fix atrivialization. For any strictly local scheme S over Spec( k ), we denote by p S : S → BG the objectof P BG corresponding to the trivial G S -torsor and by a S : S → Spec( k ) the projection. By thedefinition of BG (1.5.2), morphisms p S → p T in P ′ BG correspond bijectively to pairs ( f, r ), where f : S → T is a morphism of schemes and r ∈ G ( S ). We denote the morphism correspondingto ( f, r ) by θ ( f, r ). If s is the closed point of S , r ( s ) ∈ G ( s ) belongs to the inverse image ofa unique connected component of G , denoted [ r ]. Let Π be the groupoid with one object andfundamental group π ( G ). The above construction defines a full functor P ′ BG → Π sending θ ( f, r )to [ r ], which induces a functor still denoted by F : P BG → Π. Since θ ( a S , r ) : p S → p Spec( k ) is in M BG and θ ( f, r ) θ ( a T ,
1) = θ ( a S , r ), θ ( f, r ) is an isomorphism in P BG . Thus P BG is a connectedgroupoid. To show that F is an equivalence of categories, it suffices to check that for all r ∈ G ( S ), θ ( a S , r ) ≡ θ ( a S , ≡ stands for equality in P BG . For this, we may assume that S is a point,say S = Spec( k ′ ). We regard r : Spec( k ′ ) → G as a geometric point of G . Since G is irreducible, X = G (1) × G G ( r ) is nonempty. Let x be a geometric point of X , and let t ∈ G ( G ) be thetautological section. Then θ ( a G (1) , t ) θ ( s ,
1) = θ ( a Spec( k ) ,
1) = θ ( a G (1) , θ ( s , , where s : Spec( k ) → G (1) is the closed point. It follows that θ ( a G (1) , t ) ≡ θ ( a G (1) , θ ( a x , t ) ≡ θ ( a x , θ ( a G ( r ) , t ) ≡ θ ( a G ( r ) , s r : Spec( k ′ ) → G ( r ) denotes the closedpoint, we have θ ( a Spec( k ′ ) , r ) = θ ( a G ( r ) , t ) θ ( s r , ≡ θ ( a G ( r ) , θ ( s r ,
1) = θ ( a Spec( k ′ ) , . Construction 7.12.
Let X be a locally noetherian Artin stack. If F is a cartesian sheaf on X ,then the presheaf p ′ F : ( a : S → X ) Γ( S, a ∗ F ) ≃ F s (where s is the closed point of S ) on P ′X defines a presheaf on P X , which will denote by p F . Wethus get an exact functor(7.12.1) p : Sh cart ( X ) → ˆ P X . If X is a Deligne-Mumford stack, then p ∗ ≃ ι ∗ p , where p : ˆ P X → X et is the projection (7.1.1) and ι : P X → P X the equivalence of Proposition 7.7.The following result generalizes Proposition 7.2. Proposition 7.13.
Let X be an Artin stack, let Λ be a noetherian commutative ring, and let F be a constructible sheaf of Λ -modules on X . Then the map (7.13.1) Γ( X , F ) → lim ←− x ∈P X F x defined by the restriction maps Γ( X , F ) → ( p F )( x ) = F x is an isomorphism. The proof will be given after a couple of lemmas.46 emma 7.14.
Let F : C → D be a functor between small categories. Assume that for any morphism f : X → Y in D , there exists a morphism a : A → B in C and a commutative square in D of thefollowing form: X f / / ≃ (cid:15) (cid:15) Y ≃ (cid:15) (cid:15) F ( A ) F ( a ) / / F ( B ) . Then F is of descent for presheaves. More precisely, for any presheaf F on D , with the notationof [50, IV 4.6], the sequence F → F ∗ F ∗ F ⇒ F ∗ F ∗ F is exact, where C × D C is the 2-fiber product, F : C × D C → D is the projection, and the doublearrow is induced by the two projections from
C × D C to C . In particular, the sequence (7.14.1) Γ( ˆ D , F ) → Γ( ˆ C , F ∗ F ) ⇒ Γ( \ C × D C , F ∗ F ) is exact.Proof. For any X in D , F ( X ) → ( F ∗ F ∗ F )( X ) ⇒ ( F ∗ F ∗ F )( X ) is (7.14.1) applied to the functor F ′ : C /X → D /X induced by F and the presheaf F | ( D /X ). Since F ′ also satisfies the assumption ofthe lemma, it suffices to prove that (7.14.1) is exact. By definition, Γ( ˆ C , F ∗ F ) consists of families s = ( s X ) ∈ lim ←− X ∈C F ( F ( X )). Similarly, Γ( \ C × D C , F ∗ F ) = lim ←− ( Y,Z,α ) ∈C× D C F ( F ( Y, Z, α )). Let E be the equalizer of the double arrow in (7.14.1). We construct ǫ : E → Γ( ˆ D , F ) as follows. Let s ∈ E . For any object X of D , put ǫ ( s ) X = F ( e )( s A ) ∈ F ( X ), for a choice of e : X ∼ −→ F ( A ).This does not depend on the choice of e , because if e ′ : X ∼ −→ F ( A ′ ), then ( A, A ′ , e ′ e − ) definesan object of C × D C , and s ∈ E implies F ( e )( s A ) = F ( e ′ )( s A ′ ). For any morphism f : X → Y in D , the hypothesis implies that F ( f )( ǫ ( s ) Y ) = ǫ ( s ) X . This finishes the construction of ǫ It isstraightforward to check that ǫ is an inverse of Γ( ˆ D , F ) → E . Lemma 7.15.
Let f : X → Y be a smooth surjective morphism of Artin stacks. If U is a universecontaining P ′X and P ′Y , then the functor P ′ f : P ′X → P ′Y satisfies the condition of Lemma 7.14 for U .Proof. Let ( h, α ) : (
S, u ) → ( T, v ) be a morphism in P ′Y . Since X × Y T is an Artin stack smoothover T , it admits a section, giving rise to the following 2-commutative diagram S h / / u + + T g / / v ❅❅❅❅❅❅❅❅ X f (cid:15) (cid:15) Y . ✎✎✎✎ C K α (cid:0)(cid:0)(cid:0)(cid:0) < D β Then the following diagram commutes(
S, u ) ( h,α ) / / (cid:15) (cid:15) ( T, v ) (cid:15) (cid:15) P ′ f (( S, gh )) P ′ f (( h, id)) / / P ′ f (( T, g )) , where the left (resp. right) vertical arrow is the isomorphism (id S , βα : u → f gh ) (resp. (id T , β : v → f g )). Proof of Proposition 7.13.
Note that lim ←− x ∈P X F x → lim ←− x ∈P ′X F x is an isomorphism by Lemma 7.3.Let f : X → X be a smooth atlas. The following diagram commutes:Γ( X , F ) (cid:15) (cid:15) / / Γ( X, f ∗ F ) / / / / (cid:15) (cid:15) Γ( X × X X, g ∗ F ) (cid:15) (cid:15) lim ←− x ∈P ′X F x / / lim ←− x ∈P ′ X F x / / / / lim ←− x ∈P ′ X ×X X F x . g : X × X X → X and the double arrows are induced by the two projections from X × X X to X . The top row is exact by the definition of a sheaf. The bottom row is exact by Lemmas 7.14,7.15 and Remark 7.10 (b). The middle and right vertical arrows are isomorphisms by Propositions7.2 and 7.7. It follows that the left vertical arrow is also an isomorphism. Example 7.16.
For X = BG as in Example 7.11, F corresponds (by Corollaries 2.5 and 2.6) toa Λ-module of finite type M equipped with an action of π ( G ). Thus Γ( BG, F ) is the moduleof invariants M π ( G ) . By Example 7.11, lim ←− x ∈P BG F x is the set of zero cycles Z ( π ( G ) , M ), and(7.13.1) is the tautological isomorphism.If G is finite, the isomorphism (7.13.1) extends to an isomorphism R Γ( BG, F ) ∼ −→ R lim ←− x ∈P BG F x = R Γ( Bπ ( G ) , M ) . However, this no longer holds for G general, as the example of G = G m and F = Λ already shows(Theorem 4.4).In the rest of this section, we fix a prime number ℓ . Definition 7.17.
Let X be an Artin stack. By an ℓ -elementary point of X we mean a representable morphism x : S → X , where S is isomorphic to a quotient stack [ S/A ], where S is a strictly localscheme endowed with an action of an elementary abelian ℓ -group A acting trivially on the closedpoint of S . If x : [ S/A ] → X , y : [ T /B ] → X are ℓ -elementary points of X , a morphism from x to y is an isomorphism class of pairs ( ϕ, α ), where ϕ : [ S/A ] → [ T /B ] is a morphism and α : x → yϕ isa 2-morphism. An isomorphism between two pairs ( ϕ, α ) → ( ψ, β ) is a 2-morphism c : ϕ → ψ suchthat β = ( y ∗ c ) ◦ α . We thus get a category C ′X ,ℓ , full subcategory of Stack rep / X (Remark 1.20). Proposition 7.18.
Let X be an Artin stack.(a) Let x : S = [ S/A ] → X be an ℓ -elementary point of X , let s be the closed point of S , and let ε be the composition s → S → S . Then Aut S ( s ) ( ε ) = A , and the morphism x induces aninjection Aut S ( s ) ( ε ) ֒ → Aut X ( s ) ( x ) . (b) Let x : [ S/A ] → X , y : [ T /B ] → X be ℓ -elementary points of X , and let ( ϕ, α ) : x → y be amorphism in C ′X ,ℓ . Then there exists a pair ( f, u ) , where u : A → B is a group monomorphismand f : S → T is a u -equivariant morphism of X -schemes, such that the morphism of X -stacks ( ϕ, α ) is induced by the morphism of groupoids ( f, u ) : ( S, A ) • → ( T, B ) • over X . If ( f, u ) is such a pair and r ∈ B , then ( f r, u ) is also such a pair. If ( f , u ) and ( f , u ) aretwo such pairs, then u = u and there exists a unique r ∈ B such that f = f r .(c) Assume that X = [ X/G ] for an algebraic space X over a base algebraic space U , endowedwith an action of a smooth group algebraic space G over U . Then every ℓ -elementary point x : [ S/A ] → [ X/G ] lifts to a morphism of U -groupoids ( x , i ) : ( S, A ) • → ( X, G ) • , where x : S → X and i : S × A → G . Moreover, in the situation of (b), if ( x , i ) , ( y , j ) , ( f, u ) are liftings of x , y , ϕ to U -groupoids, respectively, then there exists a unique 2-morphism of U -groupoids (Proposition 1.2) lifting α (7.18.1) ( S, A ) • ( x ,i ) % % ❏❏❏❏❏❏❏❏❏ ( f,u ) / / ( T, B ) • ( y ,j ) (cid:15) (cid:15) ( X, G ) • ✡✡✡✡ A I given by r : S → G satisfying x ( z ) = ( y f )( z ) r ( z ) and i ( z, a ) = r ( z ) − j ( f ( z ) , u ( a )) r ( za ) .Proof. (a) The first assertion follows from the definition of [ S/A ] (Notation 1.5), and the secondone from the assumption that x is representable, hence faithful.(b) Applying Proposition 1.19 to the groupoids ( S, A ) • and ( T, B ) • over X , we get a pair ( f, u ),with u : S × A → B given by Proposition 1.2 (a), such that [ f /u ] = ( ϕ, α ). The morphism u isconstant on S , hence induced by a homomorphism, still denoted u , from A to B . Since ϕ is48epresentable, u is a monomorphism. Such a pair ( f, u ) is unique up to a unique 2-isomorphism.If ( f , u ) and ( f , u ) are two choices, a 2-isomorphism from ( f , u ) • to ( f , u ) • is given by r : S → B (Proposition 1.2 (b)), which is necessarily constant, of value denoted again r ∈ B . Thenwe have f = f r and ru = u r , hence u = u .(c) The existence of the liftings follows from Proposition 1.19 applied to the three groupoids.The description of the morphisms and the 2-morphism of groupoids comes from Proposition 1.2(b). Remark 7.19.
As the referee points out, Definition 7.17 is related to the ℓ -torsion inertia stack I ( X , ℓ ) considered in [1, Proposition 3.1.3]. Indeed, P ′ I ( X ,ℓ ) can be identified with the subcategoryof C ′X ,ℓ spanned by ℓ -elementary points of the form S × BA → X and morphisms inducing id A ,where A = Z /ℓ . Definition 7.20.
For a stack of the form S = [ S/A ] as in Definition 7.17, the group A is, in viewof Proposition 7.18 (a), uniquely determined by S (up to an isomorphism). We define the rank of S to be the rank of A , and for an ℓ -elementary point x : S → X , we define the rank of x to bethe rank of S . ℓ -elementary points of rank zero are just geometric points (Definition 7.6). The fullsubcategory of C ′X (Definition 7.17) spanned by ℓ -elementary points of rank zero is the category P ′X (Definition 7.6). Definition 7.21.
We define the category of ℓ -elementary points of X to be the category(7.21.1) C X ,ℓ = N − X ,ℓ C ′X ,ℓ deduced from C ′X ,ℓ by inverting the set N X ,ℓ of morphisms given by pairs ( f, u ) (Proposition 7.18(b)) such that f : S → T carries the closed point of S to the closed point of T and u is a groupisomorphism. When no ambiguity can arise, we will remove the subscript ℓ from the notation.Although C ′X ,ℓ is only a U -category, we will see that C X ,ℓ is essentially small if X is a Deligne-Mumford stack of finite inertia or a global quotient stack (Proposition 7.26 and Remark 8.10).We may interpret C X ,ℓ with the help of an auxiliary category ¯ C ′X ,ℓ as follows. Construction 7.22.
Objects of ¯ C ′X ,ℓ are pairs ( x, A ) such that x : S → X is a geometric pointof X , A is an elementary abelian ℓ -group acting on x by X -automorphisms with trivial actionon the closed point of S , and the morphism [ S/A ] → X is representable. Morphisms of ¯ C ′X ,ℓ are pairs ( f, u ) : ( x, A ) → ( y, B ), where u : A → B is a homomorphism and f : x → y is anequivariant morphism in P ′X . Note that u is necessarily a monomorphism. By definition, ¯ C ′X ,ℓ is afull subcategory of Eq(Stack rep / X ).We have a natural functor ρ ′ : ¯ C ′X ,ℓ → C ′X ,ℓ sending ( x, A ) to [ S/A ] → X . By Proposition 7.18(a) and (b), the functor is full and essentially surjective, and in particular cofinal (Lemma 6.2). If ̟ ′ : P ′X ֒ → C ′X ,ℓ is the inclusion functor, and ¯ ̟ ′ : P ′X → ¯ C ′X ,ℓ is the functor sending x to ( x, { } ),which is also fully faithful, we have a 2-commutative diagram(7.22.1) ¯ C ′X ,ℓρ ′ (cid:15) (cid:15) P ′X ̟ ′ / / ¯ ̟ ′ = = ④④④④④④④④ C ′X ,ℓ . Let(7.22.2) ¯ C X ,ℓ = ¯ N − X ,ℓ ¯ C ′X ,ℓ be the category deduced from ¯ C ′X ,ℓ by inverting the set ¯ N X ,ℓ of morphisms ( f, u ) : ( S, A ) → ( T, B )such that f sends the closed point s of S to the closed point t of T and u : A → B is an isomorphism.The diagram (7.22.1) induces a diagram(7.22.3) ¯ C X ,ℓρ (cid:15) (cid:15) P X ̟ / / ¯ ̟ = = ③③③③③③③③ C X ,ℓ . ρ is essentially surjective, and its effects on morphisms can be described as follows.Let ( x, A ) and ( y, B ) be objects of ¯ C X . The action of B on ( y, B ) by automorphisms in ¯ C ′X induces an action of B on ( y, B ) by automorphisms in ¯ C X , and, in turn, an action of B onHom ¯ C X (( x, A ) , ( y, B )). This action is compatible with composition in the sense that if f : ( x, A ) → ( y, B ), g : ( y, B ) → ( z, C ) are morphisms of ¯ C X , and b ∈ B , then g ◦ ( f b ) = ( g ( θ ( g )( b ))) ◦ f , where θ : ¯ C X → A is the functor induced by the functor ¯ C ′X → A carrying ( x, A ) to A . Here A denotes thecategory whose objects are elementary abelian ℓ -groups and whose morphisms are monomorphisms. Proposition 7.23. (a) The functor ρ induces a bijection (7.23.1) Hom ¯ C X (( x, A ) , ( y, B )) /B ∼ −→ Hom C X ( ρ ( x, A ) , ρ ( y, B )) . (b) The functors ¯ ̟ and ̟ are fully faithful.Proof. (a) Indeed, consider the quotient category C ♯ X having the same objects as ¯ C X with morphismsdefined by Hom C ♯ X (( x, A ) , ( y, B )) = Hom ¯ C X (( x, A ) , ( y, B )) /B, and the quotient functor ρ ♯ : ¯ C X → C ♯ X . By the universal properties of ρ , ρ ♯ , and the localizationfunctors ¯ C ′X → ¯ C X , C ′X → C X , we obtain an equivalence between C ♯ X and C X , compatible with ρ and ρ ♯ .(b) It follows from (a) that ρ induces an equivalence of categories from the full subcategory of¯ C X spanned by the image of P X to the full subcategory of C X spanned by the image of P X . Thusit suffices to show that ¯ ̟ is fully faithful. The functor ¯ C ′X → P ′X sending ( x, A ) to x is a quasi-retraction of ¯ ̟ ′ , and induces a quasi-retraction of ¯ ̟ . Here, by a quasi-retraction of a functor F , wemean a functor G endowed with a natural isomorphism GF ≃ id. Thus ¯ ̟ is faithful. Let us showthat ¯ ̟ is full. Let x, x ′ be geometric points of X . By definition, any morphism f : x → x ′ in ¯ C X isof the form ( t n , v n )( s n , u n ) − . . . ( t , v )( s , u ) − , where ( t i , v i ) : ( x i , A i ) → ( y i +1 , B i +1 ) is in ¯ C ′X and ( s i , u i ) : ( x i , A i ) → ( y i , B i ) is in ¯ N X for 1 ≤ i ≤ n , y = x , y n +1 = x ′ , B = B n +1 = { } . Then u i : A i → B i is an isomorphism and v i : A i → B i +1 is a monomorphism. Thus A i = B i = { } .Moreover, t i is in P ′X and s i is in M X . It follows that f = ¯ ̟ ( a ), where a = t n s − n . . . t s − is in P X . Remark 7.24.
For any representable morphism f : X → Y of Artin stacks, composition with f induces functors C f : C X → C Y and ¯ C f : ¯ C X → ¯ C Y . As in Remark 7.10 (a), C f and ¯ C f areequivalences of categories if f is a schematic universal homeomorphism. Definition 7.25.
Morphisms in the categories ¯ C X ,ℓ and C X ,ℓ are in general difficult to describe.When X is a Deligne-Mumford stack of finite inertia, the categories ¯ C X ,ℓ and C X ,ℓ admit simplerdescriptions, as in Proposition 7.7. Let us call a DM ℓ -elementary point of X a pair ( x, A ), where x : s → X is a geometric point of X and A an ℓ -elementary abelian subgroup of Aut X ( s ) ( x ). Definea morphism from ( x : s → X , A ) to ( y : t → X , B ) to be an X -morphism X ( x ) → X ( y ) such that f ( A ) ⊂ B , where f : Aut X ( s ) ( x ) → Aut X ( t ) ( y ) is defined as follows. Note that I ( y ) := I X × X X ( y ) is finite and unramified over X ( y ) , thus is a finite disjoint union of closed subschemes of X ( y ) by[22, Corollaire 18.4.7]. For a ∈ Aut X ( s ) ( x ), the point s → I ( y ) given by a lies in same componentas the point t → I ( y ) given by f ( a ). We thus get a category ¯ C X ,ℓ . We define the category of DM ℓ -elementary points of X to be the category C X ,ℓ having the same objects as ¯ C X ,ℓ and such thatHom C X ,ℓ (( x, A ) , ( y, B )) = Hom ¯ C X ,ℓ (( x, A ) , ( y, B )) /B . We omit the subscript ℓ from the notationwhen no ambiguity arises.Note that for ( x, A ) ∈ ¯ C X , the morphism [ X ( x ) /A ] → X is representable. Proposition 7.26.
Let X be a Deligne-Mumford stack of finite inertia. Then the functor C X → ¯ C ′X carrying ( x, A ) to ( X ( x ) → X , A ) induces an equivalence of categories ι : ¯ C X → ¯ C X and, in turn,an equivalence of categories C X → C X .Proof. A quasi-inverse of ι is induced by the functor ¯ C ′X → ¯ C X carrying ( S → X , A ) to ( s → X , A ),where s is the closed point of S . 50n the sequel, for X a Deligne-Mumford stack of finite inertia, we will often identify the cat-egories C X and C X by the equivalence of Proposition 7.26 and call DM ℓ -elementary points just ℓ -elementary points. Construction 7.27.
Let X be an Artin stack. Let F be a cartesian sheaf on X . If x : [ S/A ] → X is an ℓ -elementary point of X , let F x := x ∗ F , andΓ( x, F x ) := Γ([ S/A ] , F x ) ≃ Γ( BA, F s ) ≃ F As . If ( ϕ, α ) : [
S/A ] → [ Y /B ] is a morphism in C ′X , we have a natural map Γ( x, F x ) → Γ( y, F y ) givenby restriction, and in this way we get a presheaf q ′ F : x Γ( x, F x ) on C ′X , which factors througha presheaf q F on C X . The canonical restriction maps Γ( X, F ) → Γ( x, F x ) yield a map(7.27.1) Γ( X, F ) → lim ←− x ∈C X Γ( x, F x ) . If x : [ S/A ] → X is an elementary point of rank zero, i.e. a geometric point of X (Definition 7.17),Γ( x, F x ) = F x , and by restriction via ̟ : P X ֒ → C X , the presheaf q F induces the presheaf p F (Construction 7.12). Therefore we have a commutative diagram(7.27.2) Γ( X, F ) ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ / / lim ←− x ∈C X Γ( x, F x ) (cid:15) (cid:15) lim ←− x ∈P X F x , where the horizontal (resp. oblique) map is (7.27.1) (resp. (7.13.1)), and the vertical one is restric-tion via ̟ . Proposition 7.28.
Let X be a locally noetherian Artin stack, Λ a noetherian commutative ring,and F a constructible sheaf of Λ -modules on X . Then (7.27.1) is an isomorphism.Proof. The oblique map of (7.27.2) is an isomorphism by Proposition 7.13. By Lemma 7.3, thevertical map is obtained by applying the functor Γ( c C ′X , − ) = lim ←− C ′X ( − ) to the adjunction map α : q ′ F → ̟ ′∗ ̟ ′∗ q ′ F = ̟ ′∗ p ′ F , where ̟ ′ : P ′X → C ′X . Thus it suffices to show that α is an isomorphism. Here( ̟ ′∗ , ̟ ′∗ ) : c P ′X → c C ′X is the morphism of topoi defined by ( ̟ ′∗ E )( z ) = E ( ̟ ′ ( z )) and ( ̟ ′∗ G )( x ) = lim ←− ( t,φ ) ∈ ( ̟ ′ ↓ x ) G t ,where for an ℓ -elementary point x : [ S/A ] → X , ( ̟ ′ ↓ x ) is the category of pairs ( t, φ ), where t is ageometric point of X and φ : ̟ ′ t → x is a morphism in C ′X , which is equivalent to P ′ [ S/A ] . Let A bethe groupoid with one object ∗ and fundamental group A . Consider the functor F : A → ( ̟ ′ ↓ x )sending ∗ to ( xε, ε ), where ε : S → [ S/A ], and a ∈ A to the morphism xε → xε induced by theaction of a . For any object ( t, φ ) of ( ̟ ′ ↓ x ), the category (( t, φ ) ↓ F ) is a simply connectedgroupoid. Therefore, F is cofinal and α ( x ) : Γ( x, F x ) → lim ←− ( t,φ ) ∈ ( ̟ ′ ↓ x ) F t ∼ −→ lim ←− A F xε ≃ F As is an isomorphism. Here s is the closed point of S .In the next section we study higher cohomological variants of (7.27.1).51 A generalization of the structure theorems to Artin stacks
In this section we fix an algebraically closed field k and a prime number ℓ invertible in k . Construction 8.1.
Let X be an Artin stack, and let K ∈ D cart ( X , F ℓ ). For q ∈ Z , consider thepresheaf of F ℓ -vector spaces on C X (7.21.1)( x : [ S/A ] → X ) H q ([ S/A ] , K x ) ≃ H q ( BA, K s )(where K x := x ∗ K and s is the closed point of S ), and let(8.1.1) R q ( X , K ) := lim ←− ( x : S→X ) ∈C X H q ( S , K x ) . The restriction maps H q ( X , K ) → H q ( S , K x ) define a map(8.1.2) a q X ,K : H q ( X , K ) → R q ( X , K ) . We denote by a X ,K the direct sum of these maps:(8.1.3) a X ,K = M q a q X ,K : H ∗ ( X , K ) → R ∗ ( X , K ) . If K has a (pseudo-)ring structure (Construction 3.8), then both sides of (8.1.3) are F ℓ -(pseudo-)algebras, and a X ,K is a homomorphism of F ℓ -(pseudo-)algebras. Definition 8.2.
We say that an Artin stack X over k is a global quotient stack if X is equivalent toa stack of the form [ X/G ] for X a separated algebraic space of finite type over k and G an algebraicgroup over k . We say that an Artin stack X of finite presentation over k has a stratification byglobal quotients if there exists a stratification of X red by locally closed substacks such that eachstratum is a global quotient stack.Recall that an Artin stack over k is of finite presentation if and only if it is quasi-separatedand of finite type over k . Note that our Definition 8.2 differs from [15, Definition 2.9] and [30,Definition 3.5.3] because we allow quotients by non-affine algebraic groups.The following theorem is our main result. Theorem 8.3.
Let X be an Artin stack of finite presentation over k admitting a stratification byglobal quotients, K ∈ D + c ( X , F ℓ ) .(a) R q ( X , K ) is a finite-dimensional F ℓ -vector space for all q . Moreover, R ∗ ( X , F ℓ ) is a finitelygenerated F ℓ -algebra and, for K in D bc ( X , F ℓ ) , R ∗ ( X , K ) is a finitely generated R ∗ ( X , F ℓ ) -module.(b) If K is a pseudo-ring in D + c ( X , F ℓ ) , then Ker a X ,K (8.1.2) is a nilpotent ideal of H ∗ ( X , K ) .If, moreover, K is commutative and X is a Deligne-Mumford stack with finite inertia or aglobal quotient stack, then a X ,K is a uniform F -isomorphism (Definition 6.10). Remark 8.4. (a) A non separated scheme of finite presentation over k is not a global quotientstack in the sense of Definition 8.2 in general. Michel Raynaud gave the example of an affineplane with doubled origin. More generally, if Y is a separated smooth scheme of finite typeover k , and Y ′ is obtained by gluing two copies of Y , Y (1) and Y (2) , along the complement ofa nonempty closed subset of codimension ≥
2, then, for any algebraic group G , every G -torsor X over Y ′ is non separated. To see this, we may assume G smooth. By étale localization on Y , we may further assume that X admits a section s i over Y ( i ) , i = 1 ,
2. Assume that X is separated. The restrictions of s and s to V = Y (1) ∩ Y (2) provide a section of G × V ,which extends by Weil’s extension theorem (see [7, Theorem 4.4.1] for a generalization) toa section of G × Y (2) . Via this section, s and s can be glued to give a trivialization of X over Y ′ , contradicting the separation assumptions.(b) Recall [30, Proposition 3.5.9] that, if for every geometric point η → X , the inertia I η = η × X I X is affine, where I X = X × ∆ X , X ×X , ∆ X X , then X has a stratification by globalquotients in the sense of [30, Definition 3.5.3], and a fortiori in the sense of Definition 8.2.52c) On the other hand, the fact that X has a stratification by global quotients in the sense ofDefinition 8.2 imposes restrictions on its inertia groups. In fact, if k has characteristic zero,then, for any geometric point η = Spec( K ) → X with K algebraically closed, I η / ( I η ) aff isan abelian variety over K defined over k . Here I η is the identity component of I η and ( I η ) aff is the largest connected affine normal subgroup of I η . Indeed, if X = [ X/G ], then I η is asubgroup of G ⊗ k K , so that I η / ( I η ) aff is isogenous to an abelian subvariety of ( G /G aff ) ⊗ k K ,hence is defined over k (for an abelian variety A over k , torsion points of order invertible in k of A ⊗ k K are defined over k as k is algebraically closed).(d) For an Artin stack X of finite presentation over k and a commutative ring K in D bc ( X , F ℓ ),we do not know whether H ∗ ( X , K ) is a finitely generated F ℓ -algebra or whether a X ,K is auniform F -isomorphism in general, even under the assumption that X has a stratificationby global quotients. It may be the case that to treat the general case we would need toreformulate the theory in a relative setting.The proof of Theorem 8.3 will be given in Section 10. In the rest of this section we show thatTheorem 8.3 (b) implies Theorem 6.17 (b). Construction 8.5.
Let G be an algebraic group over k and X an algebraic space over k endowedwith an action of G (here we do not assume X to be of finite type over k ). To show that Theorem8.3 (b) implies Theorem 6.17 (b), we will proceed in two steps.(1) For K ∈ D +cart ([ X/G ] , F ℓ ) we will construct a homomorphism(8.5.1) α : R ∗ ([ X/G ] , K ) → R ∗ G ( X, K ) , which will be a homomorphism of F ℓ -(pseudo-)algebras if K has a (pseudo-)ring structure,and whose composition with a [ X/G ] ,K : H ∗ ([ X/G ] , K ) → R ∗ ([ X/G ] , K ) will be a G ( X, K )(6.16.7).(2) We will show that α is an isomorphism.Let us construct α . Recall that R q ([ X/G ] , K ) = lim ←− ( x : S→ [ X/G ]) ∈C [ X/G ] H q ( S , K x ) , and R qG ( X, K ) = lim ←− ( A,A ′ ,g ) ∈A G ( k ) ♮ H ( X A ′ , R q π ∗ [1 /c g ] ∗ K ) (6.16.8). We first compare the cat-egories A G ( k ) ♮ and C [ X/G ] by means of a third category C X,G mapping to them by functors E and Π:(8.5.2) C X,GE { { ✇✇✇✇✇✇✇✇✇ Π $ $ ❍❍❍❍❍❍❍❍❍ C [ X/G ] A G ( k ) ♮ . The category C X,G is cofibered over A G ( k ) ♮ by Π. The fiber category of C X,G at an object (
A, A ′ , g )of A G ( k ) ♮ is the category of points P X A ′ of the fixed point space of A ′ in X . If ( a, b ) : ( A, A ′ , g ) → ( Z, Z ′ , h ) is a morphism in A G ( k ) ♮ (cf. (6.16.2)), we define the pushout functor P b − : P X A ′ → P X Z ′ to be the functor induced by b − : X A ′ → ( X A ′ ) b − = X bA ′ b − ⊂ X Z ′ . If x : s → X A ′ is ageometric point of X A ′ , let E ( A,A ′ ,g ) ( x ) : [ s/A ] → [ X/G ] be the ℓ -elementary point of [ X/G ] definedby the composition E ( A,A ′ ,g ) ( x ) : [ s/A ] [ x/A ] −−−→ [ X A ′ /A ] [1 /c g ] −−−−→ [ X/G ] . For ( x : s → X A ′ ) ∈ P X A ′ , ( y : t → X Z ′ ) ∈ P X Z ′ , let u : x → y be a morphism in C X,G above( a, b ) : (
A, A ′ , g ) → ( Z, Z ′ , h ). The morphism E ( u ) : E ( A,A ′ ,g ) ( x ) → E ( Z,Z ′ ,h ) ( y )53s defined as follows. By definition, u is a commutative square X A ′ b − (cid:15) (cid:15) ( X A ′ ) ( x ) f (cid:15) (cid:15) x o o X Z ′ ( X Z ′ ) ( y ) , y o o where the horizontal arrows denote by abuse of notation the morphisms induced by strict localiza-tions. It gives the (2-commutative) square on the right of the diagram(8.5.3) [ X A ′ /A ] [1 /c g ] z z ✉✉✉✉✉✉✉✉✉✉ [ b − /c a ] (cid:15) (cid:15) [( X A ′ ) ( x ) /A ] [ x/A ] o o [ f/c a ] (cid:15) (cid:15) [ X/G ] ✺✺✺✺ (cid:22) (cid:30) [ X Z ′ /Z ] [1 /c h ] o o [( X Z ′ ) ( y ) /Z ] , [ y/Z ] o o whose composition with the 2-morphism (given by b ) in the left triangle of (8.5.3) (appearing in(6.16.3)) is the morphism E ( u ). This defines the functor E in (8.5.2).Fix q ∈ Z . Denote by H q ( K • ) the projective system (( ξ : S → [ X/G ]) H q ( S , K ξ )) on C [ X/G ] ,whose projective limit is R q ([ X/G ] , K ) (8.1.1). In other words, R q ([ X/G ] , K ) = Γ( \ C [ X/G ] , H q ( K • )).We have an inverse image map(8.5.4) Γ( \ C [ X/G ] , H q ( K • )) → Γ( [ C X,G , E ∗ H q ( K • )) ≃ Γ( \ A G ( k ) ♮ , Π ∗ E ∗ H q ( K • )) . By the cofinality lemma (Lemma 8.6) below,(Π ∗ E ∗ H q ( K • )) ( A,A ′ ,g ) ≃ lim ←− x ∈ P XA ′ H q ([ x/A ] , K x ) . By Proposition 7.2 (applied to the algebraic space X A ′ ), we have a natural isomorphismlim ←− x ∈ P XA ′ H q ([ x/A ] , K x ) ∼ −→ H ( X A ′ , R q π ∗ ([1 /c g ] ∗ K ))where π : [ X A ′ /A ] = BA × X A ′ → X A ′ is the projection, and [1 /c g ] : [ X A ′ /A ] → [ X/G ] is themorphism in (6.16.3). Finally, we find a natural isomorphismΓ( \ A G ( k ) ♮ , Π ∗ E ∗ H q ( K • )) ∼ −→ lim ←− ( A,A ′ ,g ) ∈A G ( k ) ♮ H ( X A ′ , R q π ∗ ([1 /c g ] ∗ K )) , which, by the definition of R qG ( X, K ) (6.16.8), can be rewritten(8.5.5) Γ( \ A G ( k ) ♮ , Π ∗ E ∗ H q ( K • )) ∼ −→ R qG ( X, K ) . The composition of (8.5.4) and (8.5.5) yields the desired map α (8.5.1). Lemma 8.6.
Let
Π :
C → E be a cofibered category, let e be an object of E , and let Π e be thefiber category of Π above e . Then the functor F : Π e → (Π ↓ e ) is cofinal. In particular, for everypresheaf F on C , (Π ∗ F )( e ) ∼ −→ lim ←− c ∈ Π e F ( c ) .Proof. For every object ( c, f : Π c → e ) of (Π ↓ e ), ( c, f ) → F ( f ∗ c ) is an initial object of (( c, f ) ↓ F ).Thus (( c, f ) ↓ F ) is connected. Proposition 8.7.
Under the assumptions of Construction 8.5, the functor E is cofinal. In par-ticular, (8.5.1) is an isomorphism. Corollary 8.8.
Theorem 8.3 (b) implies Theorem 6.17 (b). roof of Proposition 8.7. The second assertion follows from the first assertion and the constructionof (8.5.1). To show the first assertion, since the functor C X,G red → C
X,G is an isomorphism andthe functor C [ X/G red ] → C [ X/G ] is an equivalence of categories by Remark 7.24, we may assume G smooth.Let N be the set of morphisms in C X,G whose image under E is an isomorphism in C [ X/G ] .Then E factors as C X,G → B := N − C X,G F −→ C [ X/G ] . By Lemma 7.3, C X,G → B is cofinal. Thus it suffices to show that F is cofinal. We will show that:(a) F is essentially surjective;(b) F is full.This will imply that F is cofinal by Lemma 6.2. For the proof it is convenient to use the followingnotation. For an object x of P X A ′ above an object ( A, A ′ , g ) of A G ( k ) ♮ , we will denote the resultingobject of B by the notation ( x, ( A, A ′ , g )) . Let us prove (a). For every ℓ -elementary point ξ : [ S/A ] → [ X/G ], we choose an algebraicclosure ¯ s of the closed point s of S and we let ¯ ξ denote the composite [¯ s/A ] → [ S/A ] ξ −→ [ X/G ].We say that a lifting σ = ( a ∈ X (¯ s ) , α ∈ H om ( A, G )(¯ s ) , ι : [ a/α ] ≃ ¯ ξ )of ¯ ξ (Proposition 7.18 (c)) is rational if α ∈ H om ( A, G )( k ). Recall that α is injective. Here H om ( A, G ) is the scheme of group homomorphisms from A to G (Section 5). A rational lifting σ of ¯ ξ defines an object ω σ = ω a,α = ( a, ( α ( A ) , α ( A ) , B and an isomorphism ψ ξ,σ : F ( ω σ ) → ¯ ξ → ξ in C [ X/G ] . By Corollary 5.2, every element of H om ( A, G )(¯ s ) is conjugate by an element of G (¯ s ) toan element of H om ( A, G )( k ). Thus every ¯ ξ admits a rational lifting. It follows that F is essentiallysurjective.Let us prove (b). For any object µ = ( x, ( A, A ′ , g )) of B , σ µ = (¯ x, c g : A → G, id) is a rationallifting of F ( µ ) and ψ F ( µ ) ,σ µ = F ( m µ ), where m µ : ω σ µ = (¯ x, ( g − Ag, g − Ag, → ( x, ( A, A ′ , g )) = µ is the inverse of the obvious morphism in N above the morphism ( g,
1) : ( g − Ag, g − Ag, ← ( A, A ′ , g ) of A G ( k ) ♮ . Now if µ and ν are objects of B and f : F ( µ ) → F ( ν ) is a morphism in C X,G ,then f = F ( m ν um − µ ), where u is obtained from the following lemma applied to f , σ = σ µ , τ = σ ν .Thus F is full. Lemma 8.9.
Let ξ : [ S/A ] → [ X/G ] , and let η : [ T /B ] → [ X/G ] be ℓ -elementary points of [ X/G ] .For every morphism f : ξ → η in C [ X/G ] , every rational lifting σ of ¯ ξ , and every rational lifting τ of ¯ η , there exists a morphism u : ω σ → ω τ in B making the following diagram commute: F ( ω σ ) F ( u ) / / ψ ξ,σ (cid:15) (cid:15) F ( ω τ ) ψ η,τ (cid:15) (cid:15) ξ f / / η. Proof.
Given a triple ( f, σ, τ ) as in the lemma, we say that L ( f, σ, τ ) holds if there exists u satisfyingthe condition of the lemma. Given f : ξ → η , we say that L ( f ) holds if for every rational lifting σ of ¯ ξ and every rational lifting τ of ¯ η , L ( f, σ, τ ) holds. Step 1. First reductions. If L ( f : ξ → η, σ, τ ) and L ( g : η → ζ, τ, κ ) hold, where σ , τ , κ arerational liftings of ξ , η , ζ , respectively, then L ( gf, σ, κ ) holds, where ( gf, σ, κ ) is the composedtriple ( gf, σ, κ ) = ( g, τ, κ )( f, σ, τ ). In particular, if L ( f : ξ → η ) and L ( g : η → ζ ) hold, then L ( gf )55olds. Moreover, if L ( f ) holds for an isomorphism f , then L ( f − ) holds. Thus we may assumethat f is a morphism of C ′ [ X/G ] . Then f = ([ h/γ ] , θ ), where ( h : S → T, γ : A → B ) is an equivariantmorphism and θ : ξ → η ′′ := η [ h/γ ] is a 2-morphism. Note that f can be decomposed as ξ f −→ η ′′ f −→ η ′ f −→ η, where η ′ = η [id T /γ ], f = (id [ S/A ] , θ ), f = ([ h/ id A ] , id η ′′ ), f = ([id T /γ ] , id η ′ ), as shown by thediagram [ S/A ] ξ ✓✓✓✓ E M θ [ S/A ] [ h/ id A ] / / η ′′ $ $ ❍❍❍❍❍❍❍❍❍ [ T /A ] [id T /γ ] / / η ′ (cid:15) (cid:15) [ T /B ] η z z ✉✉✉✉✉✉✉✉✉ [ X/G ] . Step 2. L ( f ) holds for any morphism of the form f = (id [ S/A ] , θ ) , and in particular L ( f ) holds. Let σ = ( a, α, ι ) and τ = ( b, β, ǫ ) be rational liftings of ¯ ξ and ¯ η , respectively. Via the liftings, θ isgiven by g ∈ J (¯ s ), where J = Trans G ( β ( A ) , α ( A )), and a = bg . Let g ′ ∈ J ( k ) be a rational pointof the connected component of J containing g . Then h := g ′− g ∈ H (¯ s ), where H is the identitycomponent of Norm G ( α ( A )). Let e be the generic point of H ¯ s . Note that P [ H ¯ s /H ¯ s ] is equivalentto P ¯ s , and hence is a simply connected groupoid. Thus the morphism in P [ H ¯ s /H ¯ s ] induced by thediagram 1 ← e → h in P H ¯ s can be identified with the 2-morphism i → i h given by h , where i , i h : ¯ s → [ H ¯ s /H ¯ s ] are the morphisms induced by 1 and h , respectively. Then we can take u tobe the morphism( a, ( α ( A ) , α ( A ) , v −→ ( bg ′ , ( α ( A ) , α ( A ) , w −→ ( b, ( β ( A ) , β ( A ) , . in B , where v is given by the diagram 1 ← e → h in P H ¯ s via the H -equivariant morphism H ¯ s → X A carrying 1 to a (and carrying h to bg ′ ), and w is the obvious morphism of C X,G above the morphism( g ′− , g ′ ) : ( α ( A ) , α ( A ) , → ( β ( A ) , β ( A ) ,
1) of A G ( k ) ♮ . Step 3. If L ( f, σ, τ ) holds for a triple ( f, σ, τ ) , then L ( f ) holds. Indeed, if σ ′ and τ ′ are rationalliftings of ¯ ξ and ¯ η , respectively, then, by Step 2, L (id ξ , σ ′ , σ ) and L (id η , τ, τ ′ ) hold, so L ( f, σ ′ , τ ′ )holds because ( f, σ ′ , τ ′ ) = (id η , τ, τ ′ )( f, σ, τ )(id ξ , σ ′ , σ ). Step 4. L ( f ) holds. Indeed, a rational lifting τ = ( b, β, ǫ ) of ¯ η induces a rational lifting of η ′ ,and with respect to these liftings we can take u to be the morphism in B induced by the diagramin C X,G ( b, ( A, A, ← ( b, ( A, B, → ( b, ( B, B, A G ( k ) ♮ ( A, A, (id A ,γ ) ←−−−− ( A, B, ( γ, id A ) −−−−→ ( B, B, . Step 5. L ( f ) holds. By Proposition 7.18 (c), η ′ can be lifted to a morphism of groupoids( b, α ), where b : T → X , and α : T × A → G is a crossed homomorphism, which restricts to ahomomorphism T A × A → G , corresponding to a morphism, denoted by α | T A , from the (strictlylocal) scheme T A to the scheme H om ( A, G ) of group homomorphisms from A to G (Section 5).We will first show that, up to replacing T A by a finite radicial extension, α | T A is conjugate toa k -rational point of H om ( A, G ). For this, recall (Corollary 5.2) that the orbits of G acting byconjugation on H om ( A, G ) form a finite cover by open and closed subschemes. Let C ⊂ H om ( A, G )be the orbit containing the image of α | T A . Choose a k -rational point α ′ ∈ H om ( A, G )( k ) of C .Then the homomorphism g c g ( α ′ ) = g − α ′ g from G onto C factors through an isomorphism H \ G ∼ −→ C, for a subgroup H of G . Let T ′ be defined by the cartesian square T ′ / / (cid:15) (cid:15) H red \ G (cid:15) (cid:15) T A α | T A / / C. G → H red \ G is smooth, the upper horizontal arrow can be lifted to a morphism g : T ′ → G . Then c g − ( α | T ′ ) : T ′ → H om ( A, G ) is the constant map of value α ′ . Let π : T ′ → T A ֒ → T be the composite. We obtain a lifting ( bπ, α ′ ) : ( T ′ , A ) → ( X, G ) of [ T ′ /A ] → [ X/G ],which induces rational liftings of η ′ and η ′′ . With respect to these liftings, we can take u to be themorphism ¯ s → ¯ t in P X A above ( A, A,
Remark 8.10.
The categories C X,G and hence N − C X,G are essentially small. It follows from (a)and (b) in the proof of Proposition 8.7 that C [ X/G ] is essentially small. The main results of this section are the Künneth formulas of Propositions 9.5 and 9.6. One mayhope for more general formulas involving derived categories of modules over derived rings. We willnot tackle this question. Instead, we use an elementary approach, based on module structures onspectral sequences, described in Construction 9.1 and Lemma 9.2.
Construction 9.1.
Let ( C , T ) be an additive category with translation. For objects M and N in C ,the extended homomorphism group is the graded abelian group Hom ∗ ( M, N ) with Hom n ( M, N ) =Hom(
M, T n N ). The extended endomorphism ring End ∗ ( M ) = Hom ∗ ( M, M ) is a graded ring andHom ∗ ( M, N ) is a (End ∗ ( N ) , End ∗ ( M ))-bimodule. Let A ∗ be a graded ring. A left A ∗ -modulestructure on an object M of C is by definition a homomorphism λ M : A ∗ → End ∗ ( M ) of gradedrings. More precisely, such a structure is given by morphisms λ a : M → T n M , a ∈ A n , n ∈ Z suchthat λ a + b = λ a + λ b for a, b ∈ A n and the diagram M λ b / / λ ab $ $ ■■■■■■■■■ T n M T n λ a (cid:15) (cid:15) T m + n M commutes for a ∈ A m , b ∈ A n . A morphism M → M ′ in C , with M and M ′ endowed with A ∗ -module structures, is said to preserve the A ∗ -module structures if it commutes with all λ a , a ∈ A n , n ∈ Z . Let B ∗ be a graded right A ∗ -module. A morphism B ∗ ⊗ A ∗ M → N is by definitiona homomorphism B ∗ → Hom ∗ ( M, N ) of right A ∗ -modules. More precisely, it is given by a familyof morphisms f b : M → T n N , b ∈ B n , n ∈ Z in C such that f b + c = f b + f c for b, c ∈ B n and thediagram M λ a / / f ba $ $ ❍❍❍❍❍❍❍❍❍ T m M T m f b (cid:15) (cid:15) T m + n N commutes for a ∈ A m , b ∈ B n . We thus get a functor N Hom( B ∗ ⊗ A ∗ M, N ) from C tothe category of abelian groups, contravariant in M . In the category of graded abelian groupswith translation given by shifting, the notion of left A ∗ -module coincides with the usual notionof graded left A ∗ -module and the above functor is represented by the usual tensor product. Let F : ( C , T ) → ( C ′ , T ) be a functor of additive categories with translation [27, Definition 10.1.1 (ii)].A left A ∗ -module structure on M induces a left A ∗ -module structure on F M and a morphism B ∗ ⊗ A ∗ M → N induces a morphism B ∗ ⊗ A ∗ F M → F N .Let D be a triangulated category, and let A be an abelian category. We consider the additivecategories of spectral objects SpOb( D ), SpOb( A ) of type ˜ Z [46, II 4.1.2, 4.1.4, 4.1.6]. Here ˜ Z is thecategory associated to the ordered set Z ∪{±∞} . For m ∈ Z , ( X, δ ) ∈ SpOb( D ), ( H, δ ) ∈ SpOb( A ),we put ( X, δ )[ m ] = ( X [ m ] , ( − m δ [ m ]) , ( H n , δ n ) n [ m ] = ( H n + m , ( − m δ n + m ) n . For a ∈ Z ∪ {∞} , let SpSeq a ( A ) be the category of spectral sequences E a ⇒ H in A . We define( E pqa ⇒ H n )[ m ] = ( E p + m,qa ⇒ H n + m )57y multiplying all d r by ( − m . We endow SpOb( D ), SpOb( A ) and SpSeq a ( A ) with the translationfunctor [1]. The resulting categories with translation are covariant in D and A for exact functors. If H : D → A is a cohomological functor, the induced functor SpOb( D ) → SpOb( A ) commutes withtranslation. For b ≥ a , the restriction functor SpSeq a ( A ) → SpSeq b ( A ) commutes with translation.Using the notation of [46, II (4.3.3.2)], we obtain a functor SpOb( A ) → SpSeq ( A ), which alsocommutes with translation. A left A ∗ -module structure on an object of SpSeq a ( A ) induces left A ∗ -module structures on H ∗ and E ∗ qr for all q ∈ Z and r ∈ [ a, ∞ ]. If we put G q H n = F n − q H n , so thatthe abutment is of the form E pq ∞ ∼ −→ gr Gq H p + q , then G q preserves the A ∗ -module structure. Thedifferentials d ∗ qr : E ∗ qr → E ∗ + r,q − r +1 r and the abutment E ∗ q ∞ ∼ −→ gr Gq H ∗ are A ∗ -linear. A morphism B ∗ ⊗ A ∗ ( E a ⇒ H ) → ( E ′ a ⇒ H ′ ) induces morphisms on E ∗ qr , H ∗ , G q H ∗ , gr Gq H ∗ , compatible with d r , abutment, the projection G q → gr Gq and the inclusions G q − H ∗ → G q H ∗ → H ∗ . Lemma 9.2.
Let H and H ′ be filtered graded abelian groups, H endowed with a left A ∗ -modulestructure. We let G denote the (increasing) filtrations. Assume that G q H ∗ = G q H ′∗ = 0 for q small enough and H n = S q ∈ Z G q H n , H ′ n = S q ∈ Z G q H ′ n for all n . Let B ∗ ⊗ A ∗ H → H ′ be amorphism such that the homomorphism B ∗ ⊗ A ∗ gr Gq H ∗ → gr Gq H ′∗ is an isomorphism for all q .Then the homomorphism B ∗ ⊗ A ∗ H ∗ → H ′∗ is an isomorphism.Proof. Since G q H ∗ = G q H ′∗ = 0 for q small enough, one shows by induction that the morphismof exact sequences B ∗ ⊗ A ∗ G q − H ∗ / / (cid:15) (cid:15) B ∗ ⊗ A ∗ G q H ∗ / / (cid:15) (cid:15) B ∗ ⊗ A ∗ gr Gq H ∗ / / ≃ (cid:15) (cid:15) / / G q − H ′∗ / / G q H ′∗ / / gr Gq H ′∗ / / −→ q ∈ Z G q H ∗ = H ∗ , lim −→ q ∈ Z G q H ′∗ = H ′∗ andthe fact that tensor product commutes with colimits. Construction 9.3.
Let(9.3.1) X ′ h / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′ g / / Y be a 2-commutative square of commutatively ringed topoi, K ∈ D ( O Y ′ ), L ∈ D ( O X ). An element s ∈ H m ( Y ′ , K ) corresponds to a morphism O Y ′ → K [ m ] in D ( O Y ′ ), and an element t ∈ H n ( X, L )corresponds to a morphism O X → L [ n ] in D ( O X ). Then Lf ′∗ s ⊗ L O X ′ Lh ∗ t : O X ′ → Lf ′∗ K ⊗ L O X ′ Lh ∗ L is a morphism in D ( O X ′ ). This defines a graded map H ∗ ( Y ′ , K ) × H ∗ ( X, L ) → H ∗ ( X ′ , Lf ′∗ K ⊗ L O X ′ Lh ∗ L )which is H ∗ ( Y, O Y )-bilinear, hence induces a homomorphism(9.3.2) H ∗ ( Y ′ , K ) ⊗ H ∗ ( Y, O Y ) H ∗ ( X, L ) → H ∗ ( X ′ , Lf ′∗ K ⊗ L O X ′ Lh ∗ L ) , which is a homomorphism of ( H ∗ ( Y ′ , O Y ′ ) , H ∗ ( X, O X ))-bimodules. Construction 9.4.
Let f : X → Y be a morphism of commutatively ringed topoi, and let L ∈ D ( O Y ), K ∈ D ( O X ). We consider the second spectral object ( L, δ ) associated to L [46, III 4.3.1,4.3.4], with L ( p, q ) = τ [ p,q − L . For s ∈ H n ( Y, O Y ) corresponding to O Y → O Y [ n ], the functor s ⊗ L O Y − induces a morphism of spectral objects ( L, δ ) → ( L, δ )[ n ]. This endows ( L, δ ) with astructure of H ∗ ( Y, O Y )-module (Construction 9.1). For t ∈ H n ( X, K ) corresponding to O X → [ n ], the functor t ⊗ L O X − induces a morphism of spectral objects Lf ∗ ( L, δ ) → K ⊗ L O X Lf ∗ ( L, δ )[ n ].This defines a morphism H ∗ ( X, K ) ⊗ H ∗ ( Y, O Y ) Lf ∗ ( L, δ ) → K ⊗ L O X Lf ∗ ( L, δ ) . Applying Rf ∗ and composing with the adjunction id D ( O Y ) → Rf ∗ Lf ∗ , we get a morphism H ∗ ( X, K ) ⊗ H ∗ ( Y, O Y ) ( L, δ ) → Rf ∗ ( K ⊗ L O X Lf ∗ ( L, δ )) . Further applying the cohomological functor H ( Y, − ), we obtain a morphism H ∗ ( X, K ) ⊗ H ∗ ( Y, O Y ) ( E ⇒ H ) → ( E ′ ⇒ H ′ ) , where the two spectral sequences are E pq = H p ( Y, H q L ) ⇒ H p + q ( Y, L ) , (9.4.1) E ′ pq = H p ( X, K ⊗ L O X Lf ∗ H q L ) ⇒ H p + q ( X, K ⊗ L O X Lf ∗ L ) . (9.4.2)By construction, the induced morphisms on E ∗ q and on H ∗ coincide with (9.3.2) for (9.3.1) givenby id f .The results of Constructions 9.3 and 9.4 have obvious analogues for Artin stacks and complexesin D cart ( − , Λ), where Λ is a commutative ring.
Proposition 9.5.
Let X ′ h / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′ g / / Y be a 2-commutative square of Artin stacks. Let K ∈ D +cart ( Y ′ , F ℓ ) and L ∈ D +cart ( X , F ℓ ) . Supposethat(a) The Leray spectral sequence for ( f, L )(9.5.1) E pq = H p ( Y , R q f ∗ L ) ⇒ H p + q ( X , L ) degenerates at E .(b) For every q , R q f ∗ L is a constant constructible F ℓ -module on Y .(c) The base change morphism BC : g ∗ Rf ∗ L → Rf ′∗ h ∗ L is an isomorphism.(d) The morphism PF f ′ : Rg ∗ ( K ⊗ Rf ′∗ h ∗ L ) → Rg ∗ Rf ′∗ ( f ′∗ K ⊗ h ∗ L ) deduced from the projectionformula morphism K ⊗ Rf ′∗ h ∗ L → Rf ′∗ ( f ′∗ K ⊗ h ∗ L ) is an isomorphism.Then the spectral sequence (of type (9.4.2) ) (9.5.2) E pq = H p ( Y ′ , K ⊗ R q f ′∗ h ∗ L ) ⇒ H p + q ( Y ′ , K ⊗ Rf ′∗ h ∗ L ) degenerates at E and the homomorphism (9.3.2)(9.5.3) H ∗ ( Y ′ , K ) ⊗ H ∗ ( Y , F ℓ ) H ∗ ( X , L ) → H ∗ ( X ′ , f ′∗ K ⊗ h ∗ L ) is an isomorphism.Proof. Take any geometric point t → Y ′ . By (b), the E -term of (9.5.1) is E pq = H p ( Y , R q f ∗ L ) ≃ H p ( Y , F ℓ ) ⊗ ( R q f ∗ L ) t . By (c), (9.5.2) is isomorphic to(9.5.4) E ′ pq = H p ( Y ′ , K ⊗ g ∗ R q f ∗ L ) ⇒ H p + q ( Y ′ , K ⊗ g ∗ Rf ∗ L ) . By (b), E ′ pq ≃ H p ( Y ′ , K ) ⊗ ( R q f ∗ L ) t . Thus the morphism H ∗ ( Y ′ , K ) ⊗ H ∗ ( Y , F ℓ ) E ∗ q → E ′∗ q
59s an isomorphism. Il then follows from (a) and Lemma 9.2 that (9.5.4) degenerates at E and thehomomorphism(9.5.5) H ∗ ( Y ′ , K ) ⊗ H ∗ ( Y , F ℓ ) H ∗ ( X , L ) → H ∗ ( Y ′ , K ⊗ g ∗ Rf ∗ L )is an isomorphism. Thus (9.5.2) degenerates at E and (9.5.3) is an isomorphism since it is thecomposition of (9.5.5) with the morphism induced by the composition Rg ∗ ( K ⊗ g ∗ Rf ∗ L ) Rg ∗ (id K ⊗ BC) −−−−−−−−−→ ∼ Rg ∗ ( K ⊗ Rf ′∗ h ∗ L ) PF f ′ −−−→ ∼ Rg ∗ Rf ′∗ ( f ′∗ K ⊗ h ∗ L ) , of the isomorphisms in (c) and (d).In the rest of this section, let k be a separably closed field of characteristic = ℓ . Proposition 9.6.
Let G be a connected algebraic group over k , and let X be an algebraic space offinite presentation over k endowed with an action of G . Let X ′ h / / f ′ (cid:15) (cid:15) [ X/G ] f (cid:15) (cid:15) Y ′ g / / BG be a 2-cartesian square of quasi-compact, quasi-separated Artin stacks, where f is the canonicalprojection. Let K ∈ D +cart ( Y ′ , F ℓ ) . Suppose that the map e : H ∗ ([ X/G ]) → H ∗ ( X ) induced by theprojection X → [ X/G ] is surjective. Then H ∗ ([ X/G ]) is a finitely generated free H ∗ ( BG ) -module,the spectral sequence E pq = H p ( Y ′ , K ⊗ R q f ′∗ F ℓ ) ⇒ H p + q ( Y ′ , K ⊗ Rf ′∗ F ℓ ) degenerates at E , and the homomorphism H ∗ ( Y ′ , K ) ⊗ H ∗ ( BG ) H ∗ ([ X/G ]) → H ∗ ( X ′ , f ′∗ K ) is an isomorphism.Proof. For the second and the third assertions, we apply Proposition 9.5. By Corollary 2.6 andgeneric base change (Remark 2.12), conditions (b) and (c) of Proposition 9.5 are satisfied. For L ∈ D +cart ([ X/G ] , F ℓ ), the diagram Rg ∗ K ⊗ Rf ∗ L PF g / / PF f ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Rg ∗ ( K ⊗ g ∗ Rf ∗ L ) BC / / Rg ∗ ( K ⊗ Rf ′∗ h ∗ L ) PF f ′ / / R ( gf ′ ) ∗ ( f ′∗ K ⊗ h ∗ L ) ≃ (cid:15) (cid:15) Rf ∗ ( f ∗ Rg ∗ K ⊗ L ) BC ′ / / Rf ∗ ( Rh ∗ f ′∗ K ⊗ L ) PF h / / R ( f h ) ∗ ( f ′∗ K ⊗ h ∗ L )commutes. Take L = F ℓ . Then generic base change (Remark 2.12) and Proposition 2.9 (d) implythat BC, BC ′ , PF f , PF g , PF h are isomorphisms, hence PF f ′ is an isomorphism as well, whichproves condition (d) of Proposition 9.5. Next we check condition (a) of Proposition 9.5. The Lerayspectral sequence for f is(9.6.1) E pq = H p ( BG, R q f ∗ F ℓ ) ⇒ H p + q ([ X/G ]) . Since R q f ∗ F ℓ is constant of value H q ( X ), we have E pq ≃ H p ( BG ) ⊗ H q ( X ). As e is an edgehomomorphism for (9.6.1), its surjectivity implies d qr = 0 for all r ≥
2. It then follows from the H ∗ ( BG )-module structure of (9.6.1) that it degenerates at E . The first assertion of Proposition9.6 then follows from the fact that H ∗ ( X ) is a finite-dimensional vector space. Proposition 9.7.
Let G = GL n,k , T be a maximal torus of G , A = Ker( − ℓ : T → T ) . Then themap H ∗ ( BA, F ℓ ) → H ∗ ( G/A, F ℓ ) induced by the projection G/A → BA is surjective. roof. Let us recall the proof on [36, page 566]. Consider the following diagram with 2-cartesiansquares (Proposition 1.11):
G/A / / (cid:15) (cid:15) G/T (cid:15) (cid:15) / / Spec k (cid:15) (cid:15) BA / / BT / / BG.
Note that the arrow BA → BT can be identified with the composition BA ∼ −→ [ X/T ] → BT , where X = A \ T , and the first morphism is an isomorphism by Corollary 1.16. The map H ∗ ( BA ) → H ∗ ( X ) induced by the projection π : X = A \ T → BA is surjective. Indeed, using Künneth formulathis reduces to the case where T has dimension 1, which follows from Lemma 9.8 below. Note that π can be identified with the composition X → [ X/T ] ≃ BA . Thus, by Proposition 9.6 applied to f : [ X/T ] → BT , the map H ∗ ( G/T ) ⊗ H ∗ ( BT ) H ∗ ( BA ) → H ∗ ( G/A )is an isomorphism. We conclude by applying the fact that H ∗ ( BT ) → H ∗ ( G/T ) is surjective(Theorem 4.4).
Lemma 9.8.
Let A be an elementary abelian ℓ -group, and let X be a connected algebraic spaceendowed with an A -action such that X is the maximal connected Galois étale cover of [ X/A ] whosegroup is an elementary abelian ℓ -group. Then the homomorphism (9.8.1) H ( BA, F ℓ ) → H ([ X/A ] , F ℓ ) induced by the projection [ X/A ] → BA is an isomorphism.Proof. For any connected Deligne-Mumford stack X , H ( X , F ℓ ) is canonically identified withHom( π ( X ) , F ℓ ), and (9.8.1) is induced by the morphism π ([ X/A ]) → π ( BA ) ≃ A. The assumption means that A is the maximal elementary abelian ℓ -quotient of π ([ X/A ]).
Proposition 9.9.
Let X be an abelian variety over k , A = X [ ℓ ] = Ker( ℓ : X → X ) . Then themap H ∗ ( BA, F ℓ ) → H ∗ ( X/A, F ℓ ) induced by the projection X/A → BA is surjective.Proof. We apply Lemma 9.8 to the morphism ℓ : X → X , which identifies the target with X/A .By Serre-Lang’s theorem [48, XI Théorème 2.1], this morphism is the maximal étale Galois coverof X by an elementary abelian ℓ -group. Thus H ( BA ) → H ( X/A ) is an isomorphism. It thensuffices to apply the fact that H ∗ ( X/A ) is the exterior algebra of H ( X/A ).
10 Proof of the structure theorem
We proceed in several steps:(1) We first prove Theorem 8.3 (b) when X is a Deligne-Mumford stack with finite inertia, andwhose inertia groups are elementary abelian ℓ -groups.(2) We prove Theorem 8.3 (b) for X a quotient stack [ X/G ].(3) For certain quotient stacks [
X/G ] we establish estimates for the powers of F annihilating thekernel and the cokernel of a G ( X, K ) (6.16.9).(4) Using (3), we prove Theorem 8.3 (b) for Deligne-Mumford stacks with finite inertia.(5) We prove Theorem 8.3 (a) and the first assertion of (b) for Artin stacks having a stratificationby global quotients.
Construction 10.1.
Let f : X → Y be a morphism of commutatively ringed topoi such that ℓ O Y = 0, K ∈ D ( X ). The Leray spectral sequence of f , E ij = H i ( Y, R j f ∗ K ) ⇒ H i + j ( X, K ) , e f,K : H ∗ ( X, K ) → H ( Y, R ∗ f ∗ K ) , which is a homomorphism of F ℓ -(pseudo-)algebras if K ∈ D ( X ) is a (pseudo-)ring. The followingcrucial lemma is similar to Quillen’s result [36, Proposition 3.2]. Lemma 10.2.
Let K be a pseudo-ring in D ( X ) . Assume that c = cd( Y ) < ∞ . Then (Ker e f,K ) c +1 =0 . Moreover, if K is commutative, then e f,K is a uniform F -isomorphism; more precisely, for b ∈ E , ∗ , we have b ℓ n ∈ Im e f,K , where n = max { c − , } .Proof. We imitate the proof of [36, Proposition 3.2] (for the case of finite cohomological dimension).We have E ij = 0 for i > c . Consider the multiplicative structure on the spectral sequence(Example 3.15). As Ker e f,K = F H ∗ ( X, K ), where F • denotes the filtration on the abutment,(Ker e f,K ) c +1 ⊂ F c +1 H ∗ ( X, K ) = 0. If K is commutative and b ∈ E , ∗ r , then the formula d r ( b ℓ ) = ℓb ℓ − d r ( b ) = 0 implies that b ℓ ∈ E , ∗ r +1 . Thus for b ∈ E , ∗ , b ℓ n ∈ E , ∗ n = E , ∗∞ = Im e f,K . Construction 10.3.
Let X be a Deligne-Mumford stack of finite presentation and finite inertiaover k . By Keel-Mori’s theorem [28] (see [40, Theorem 6.12] for a generalization), there exists acoarse moduli space morphism f : X → Y, which is proper and quasi-finite. Let K ∈ D +cart ( X , F ℓ ). Then Construction 10.1 and Lemma 10.2apply to f and K with cd ℓ ( Y ) ≤ Y ).For any geometric point t of Y , consider the following diagram of Artin stacks with 2-cartesiansquares: X t / / (cid:15) (cid:15) X ( t ) / / (cid:15) (cid:15) X f (cid:15) (cid:15) t / / Y ( t ) / / Y. We have canonical isomorphisms(10.3.1) ( R q f ∗ K ) t ∼ −→ H q ( X ( t ) , K ) ∼ −→ H q ( X t , K ) , the second one by the proper base change theorem (cf. [34, Theorem 9.14]). Therefore, if we let P Y denote the category of geometric points of Y (Definition 7.1), the map(10.3.2) H ( Y, R q f ∗ K ) → lim ←− t ∈ P Y H q ( X ( t ) , K ) ∼ −→ lim ←− t ∈ P Y H q ( X t , K ) , is an isomorphism if K ∈ D + c ( X , F ℓ ), by Proposition 7.2. On the other hand, recall (8.1.1) that R q ( X , K ) = lim ←− ( x : S→X ) ∈C X H q ( S , K x ) = Γ( c C X , H q ( K • )) , where K x = x ∗ K and H q ( K • ) denotes the presheaf on C X whose value at x is H q ( S , K x ). Wedefine a category C f and functors C f ψ ❆❆❆❆❆❆❆❆ ϕ ~ ~ ⑥⑥⑥⑥⑥⑥⑥ C X P Y as follows. The category C f is cofibered over P Y by ψ . The fiber category of ψ at a geometricpoint t → Y is C X ( t ) . The pushout functor C X ( t ) → C X ( z ) for a morphism of geometric points t → z is induced by the morphism X ( t ) → X ( z ) (Remark 7.24). The functors ϕ t : C X ( t ) → C X induced bythe morphisms X ( t ) → X define ϕ . Thus we have an inverse image map(10.3.3) ϕ ∗ : R q ( X , K ) → Γ( c C f , ϕ ∗ H q ( K • )) .
62y Lemma 8.6 we have ψ ∗ ϕ ∗ H q ( K • ) t ≃ Γ( [ C X ( t ) , ϕ ∗ t H q ( K • )) . Thus we have(10.3.4) Γ( c C f , ϕ ∗ H q ( K • )) ≃ Γ( c P Y , ψ ∗ ϕ ∗ H q ( K • )) ∼ −→ lim ←− t ∈ P Y lim ←− ( x : S→X ( t ) ) ∈C X ( t ) H q ( S , K x ) Proposition 10.4. (a) The following diagram commutes H q ( X , K ) e qf,K / / a q X ,K (cid:15) (cid:15) H ( Y, R q f ∗ K ) (10.3.2) / / lim ←− t ∈ P Y H q ( X ( t ) , K ) lim ←− t ∈ PY a q X ( t ) ,K (cid:15) (cid:15) R q ( X , K ) ϕ ∗ / / Γ( c C f , ϕ ∗ H q ( K • )) (10.3.4) ∼ / / lim ←− t ∈ P Y lim ←− ( x : S→X ( t ) ) ∈C X ( t ) H q ( S , K x ) . (b) ϕ ∗ is an isomorphism.(c) Consider the commutative square H q ( X ( t ) , K ) a q X ( t ) ,K (cid:15) (cid:15) ∼ / / H q ( X t , K ) a q X t,K (cid:15) (cid:15) lim ←− S∈C X ( t ) H q ( S , K ) ι ∗ / / lim ←− S∈C X t H q ( S , K ) defined by the functor ι : C X t → C X ( t ) induced by the inclusion X t → X T , in which the upperhorizontal map is the second isomorphism of (10.3.1) . The map ι ∗ is an isomorphism.Proof. Assertion (a) follows from the definitions. For (b) it suffices to show that ϕ is cofinal. Let τ : C X → C f be the functor carrying an ℓ -elementary point x : [ S/A ] → X , with s the closed pointof S , to the induced ℓ -elementary point τ ( x ) : [ S/A ] → X ( f ( s )) . Then we have ϕτ ≃ id C X , anda canonical natural transformation τ ϕ → id C f , carrying an object ξ : [ S/A ] → X ( t ) of C f to thecocartesian morphism τ ϕ ( ξ ) → ξ in C f above the morphism f ( s ) → t in P Y . These exhibit τ asa left adjoint to ϕ . Therefore, by Lemma 10.5 below, ϕ is cofinal. For (c), it suffices again toshow that ι is cofinal. Let X → X ( t ) be an étale atlas. As f is quasi-finite, up to replacing X bya connected component, we may assume that X is a strictly local scheme, finite over Y ( t ) . Then X ( t ) ≃ [ X/G ], where G = Aut X ( t ) ( x ), x is the closed point of X . Let ξ : [ S/A ] → [ X/G ] be an ℓ -elementary point of [ X/G ]. The ℓ -elementary point [ x/A ] → X t , endowed with the morphism in C [ X/G ] given by the diagram [ S/A ] → [ X/A ] ← [ x/A ]in C ′ [ X/G ] , defines an initial object of ( ξ ↓ ι ). Therefore, ι is cofinal. Lemma 10.5.
Let G : A → B be a functor. If G has a left adjoint, then G is cofinal.Proof. Let F : B → A be a left adjoint to G . Then, for every object b of B , ( F b, b → GF b ) is aninitial object of ( b ↓ G ). Thus ( b ↓ G ) is connected. Corollary 10.6.
The assertion of Theorem 8.3 (b) holds if X is a Deligne-Mumford stack withfinite inertia, whose inertia groups are elementary abelian ℓ -groups. More precisely, if c = cd ℓ ( Y ) ,where Y is the coarse moduli space of X , then (Ker a X ,K ) c +1 = 0 and for K commutative and b ∈ E , ∗ , we have b ℓ n ∈ Im a X ,K , where n = max { c − , } .Proof. It suffices to show that, for all t ∈ P Y , a q X t ,K : H q ( X t , K ) → lim ←− ( x : S→X t ) ∈C X t H q ( S , K x )63s an isomorphism. Indeed, by Proposition 10.4 (c) this will imply that the right vertical arrowin the diagram of Proposition 10.4 (a) is an isomorphism. As (10.3.2) is an isomorphism, ϕ ∗ is an isomorphism (Proposition 10.4 (b)), and e f,K = L e qf,K has nilpotent kernel and, if K iscommutative, is an F -isomorphism (Lemma 10.2), it will follow that a X ,K = L a q X ,K has thesame properties with the same bounds for the exponents. As f : X → Y is a coarse moduli spacemorphism, there exists a finite radicial extension t ′ → t and a geometric point y ′ of X above t ′ suchthat ( X t ′ ) red ≃ B Aut X ( y ′ ). Therefore we are reduced to showing that a X ,K is an isomorphismfor X = BA k , where A is an elementary abelian ℓ -group. In this case, id BA k : BA k → BA k isa final object of C BA k , so we can identify R q ( BA k , K ) with H q ( BA k , K ), and a qBA k ,K with theidentity. Corollary 10.7.
Suppose X = [ X/G ] is a global quotient stack (Definition 8.2), where the actionof G on X satisfies the following two properties:(a) The morphism γ : G × X → X × X , ( g, x ) ( x, xg ) is finite and unramified.(b) All the inertia groups of G are elementary abelian ℓ -groups.Then the assertions of Corollary 10.6 hold.Proof. As γ in (a) can be identified with the morphism X × [ X/G ] X → X × X , which is the pull-back of the diagonal morphism ∆ [ X/G ] : [ X/G ] → [ X/G ] × [ X/G ] by X × X → [ X/G ] × [ X/G ],(a) implies that ∆ [ X/G ] is finite and unramified. In particular, [ X/G ] is a Deligne-Mumford stack.Moreover, as the inertia stack is the pull-back of ∆ [ X/G ] by ∆ [ X/G ] , [ X/G ] has finite inertia. Taking(b) into account, we see that [
X/G ] satisfies the assumptions of 10.6, and therefore 8.3 (b) holdsfor [
X/G ]. Proposition 10.8.
Theorem 8.3 (b) for global quotient stacks [ X/G ] (Definition 8.2) follows fromTheorem 8.3 (b) for G linear.Proof. Consider the system of subgroups G i = L · A [ mℓ i ] · F of G = L · A · F as in the proof ofTheorem 4.6 (with Λ = F ℓ and n = ℓ ), where m is the order of F . Note that every elementaryabelian ℓ -subgroup of A · F is contained in A [ mℓ ] · F . As a consequence, every elementary abelian ℓ -subgroup of G is contained in G , so that the restriction map R ∗ G ( X, K ) → R ∗ G i ( X, K ) is anisomorphism for i ≥
1. Consider the commutative diagram H ∗ ([ X/G ] , K ) / / a G ( X,K ) (cid:15) (cid:15) H ∗ ([ X/G d ] , K ) a G d ( X,K ) (cid:15) (cid:15) α / / H ∗ ([ X/G d ] , K ) a G d ( X,K ) (cid:15) (cid:15) R ∗ G ( X, K ) ∼ / / R ∗ G d ( X, K ) ∼ / / R ∗ G d ( X, K ) , where d = dim A . By Remark 4.9, H ∗ ([ X/G ] , K ) is the image of α . Thus it suffices to show that a G d ( X, K ) has nilpotent kernel and, if K is commutative, a G d ( X, K ) is a uniform F -surjection. Proposition 10.9.
Theorem 8.3 (b) holds for global quotient stacks of the form [ X/G ] , where G is either a linear algebraic group, or an abelian variety.Proof. Although by Proposition 10.8 it would suffice to treat the case where G is linear, we prefer totreat both cases simultaneously, in order to later get better bounds for the power of F annihilatingthe kernel and the cokernel of the map a X ,K (Corollary 10.10). We follow closely the arguments ofQuillen for the proof of [36, Theorem 6.2]. If G is linear, choose an embedding of G into a lineargroup L = GL n over k [13, Corollaire II.2.3.4], and a maximal torus T of L . If G is an abelianvariety, let L = T = G . In both cases, denote by S the kernel of ℓ : T → T , which is an elementaryabelian ℓ -group of order n . We let L act on F = S \ L by right multiplication. If g ∈ L ( k ), and if { S } denotes the rational point of F defined by the coset S , the inertia group of L at { S } g is g − Sg .Let us show that the diagonal action of G on X × F (resp. X × F × F ) satisfies assumptions (a)and (b) of Corollary 10.7. It suffices to show this for X × F . Consider the commutative square L × L (cid:15) (cid:15) ∼ / / L × L (cid:15) (cid:15) F × L / / F × F γ : ( x, g ) ( x, xg ). As the vertical morphismsare finite and surjective, so is the lower horizontal morphism. Moreover, the latter is unramified.Hence the morphism γ : F × G → F × F is finite and unramified. The same holds for the morphism γ : ( X × F ) × G → ( X × F ) × ( X × F ), ( x, y, g ) ( x, y, xg, yg ), because it is the composite X × F × G → X × X × F × G → X × F × X × F , where the first morphism ( x, y, g ) ( x, xg, y, g ) isa closed immersion by the assumption that X is separated and the second morphism ( x, x ′ , y, g ) ( x, y, x ′ , yg ) is a base change of F × G → F × F . So (a) is satisfied for X × F . Moreover, theinertia groups of G on X × F are conjugate in L to subgroups of S , so (b) is satisfied for X × F .As in [36, 6.2], consider the following commutative diagram(10.9.1) H ∗ ([ X/G ] , K ) / / a G ( X,K ) (cid:15) (cid:15) H ∗ ([ X × F/G ] , [pr / id G ] ∗ K ) / / / / a G ( X × F, [pr / id G ] ∗ K ) (cid:15) (cid:15) H ∗ ([ X × F × F/G ] , [pr / id G ] ∗ K ) a G ( X × F × F, [pr / id G ] ∗ K ) (cid:15) (cid:15) R ∗ G ( X, K ) / / R ∗ G ( X × F, [pr / id G ] ∗ K ) / / / / R ∗ G ( X × F × F, [pr / id G ] ∗ K ) , in which the double horizontal arrows are defined by pr and pr . By Corollary 10.7, a G ( X × F, [pr / id G ] ∗ K ) and a G ( X × F × F, [pr / id G ] ∗ K ) have nilpotent kernels and, if K is commutative,are uniform F -surjections. To show that a G ( X, K ) has the same properties it thus suffices to showthat the rows of (10.9.1) are exact.First consider the lower row. The component of degree q is isomorphic by definition (6.16.8)to the projective limit over ( A, A ′ , g ) ∈ A G ( k ) ♮ of(10.9.2) Γ( X A ′ , R q π ∗ r ∗ K ) → Γ( X A ′ × F A ′ , R q π ∗ r ∗ [pr / id G ] ∗ K ) ⇒ Γ( X A ′ × F A ′ × F A ′ , R q π ∗ r ∗ [pr / id G ] ∗ K ) , where we have put r := [1 /c g ]. In order to identify the second and third terms of (10.9.2), considerthe following commutative diagram, where the middle and right squares are cartesian:[ X × F/G ] [pr / id G ] (cid:15) (cid:15) BA × X A ′ × F A ′ id × pr (cid:15) (cid:15) r o o π / / X A ′ × F A ′ pr (cid:15) (cid:15) pr / / F A ′ (cid:15) (cid:15) [ X/G ] BA × X A ′ r o o π / / X A ′ / / Spec k .
We have (by base change for the middle square)pr ∗ R q π ∗ ( r ∗ K ) ∼ −→ R q π ∗ (id × pr ) ∗ r ∗ K ≃ R q π ∗ r ∗ [pr / id G ] ∗ K. By the Künneth formula for the right square, we haveΓ( X A ′ × F A ′ , pr ∗ R q π ∗ r ∗ K ) ∼ −→ Γ( X A ′ , R q π ∗ r ∗ K ) ⊗ Γ( F A ′ , F ℓ ) . Therefore we get a canonical isomorphismΓ( X A ′ × F A ′ , R q π ∗ r ∗ [pr / id G ] ∗ K ) ∼ −→ Γ( X A ′ , R q π ∗ r ∗ K ) ⊗ Γ( F A ′ , F ℓ ) . We have a similar identification for X A ′ × F A ′ × F A ′ , and these identifications produce an isomor-phism between (10.9.2) and the tensor product of Γ( X A ′ , R q π ∗ r ∗ K ) with(10.9.3) Γ(Spec k, F ℓ ) → Γ( F A ′ , F ℓ ) ⇒ Γ( F A ′ × F A ′ , F ℓ ) . As A ′ is an elementary abelian ℓ -subgroup of G , A ′ is conjugate in L to a subgroup of S , hence F A ′ = ∅ . It follows that (10.9.3), (10.9.2) and hence the lower row of (10.9.1) are exact.In order to prove the exactness of the upper row of (10.9.1), consider the square of Artin stackswith representable morphisms,(10.9.4) [( Y × F ) /G ] / / (cid:15) (cid:15) [ Y /G ] (cid:15) (cid:15) [ F/L ] / / BL, Y is an algebraic space of finite presentation over k endowed with an action of G , thehorizontal morphisms are induced by projection from F and the vertical morphisms are inducedby the embedding G → L . The square is 2-cartesian by Proposition 1.11 and BS ≃ [( S \ L ) /L ] =[ F/L ]. By Propositions 9.6, 9.7 and 9.9, H ∗ ([ F/L ]) is a finitely generated free H ∗ ( BL )-moduleand the homomorphism H ∗ ([ Y /G ] , K ) ⊗ H ∗ ( BL ) H ∗ ([ F/L ]) → H ∗ ([ Y × F/G ] , [pr / id G ] ∗ K )defined by (10.9.4) is an isomorphism. Applying the above to Y = X and Y = X × F , we obtainan identification of the upper row of (10.9.1) with the sequence H ∗ ([ Y /G ] , K ) → H ∗ ([ Y /G ] , K ) ⊗ H ∗ ( BL ) H ∗ ([ F/L ]) ⇒ H ∗ ([ Y /G ] , K ) ⊗ H ∗ ( BL ) H ∗ ([ F/L ]) ⊗ H ∗ ( BL ) H ∗ ([ F/L ]) , which is exact by the usual argument of faithfully flat descent. Corollary 10.10.
Let X = [ X/G ] be a global quotient stack, and assume that either (a) G isembedded in L = GL n , n ≥ , or (b) G is an abelian variety. Let K ∈ D + c ([ X/G ] , F ℓ ) be a pseudo-ring. Let d = dim X . In case (a), let e = dim L/G , f = 2 dim L − dim G . In case (b), let e = 0 , f = dim G . Then(i) (Ker a G ( X, K )) m = 0 , where m = 2 d + 2 e + 1 ,(ii) for K commutative and y ∈ R ∗ G ( X, K ) , we have y ℓ N ∈ Im a G ( X, K ) for N ≥ max { d + 2 e − , } + log ℓ (2 d + 2 f + 1) .Proof. As in the proof of Proposition 10.9, let F = S \ L . We have cd ℓ (( X × F ) /G ) ≤ X × F ) /G ) = 2 d + 2 e . As all inertia groups of G acting on X × F are elementary abelian ℓ -groups,by Corollary 10.7 we have (Ker a G ( X × F, pr ∗ K )) m = 0, hence (i) by (10.9.1). For (ii), set a G ( X, K ) = a , a G ( X × F, pr ∗ K ) = a , a G ( X × F × F, pr ∗ K ) = a . Denote by u : H ∗ ([ X/G ] , K ) → H ∗ ([ X × F/G ] , [pr / id G ] ∗ K ) (resp. v : R ∗ G ( X, K ) → R ∗ G ( X × F, [pr / id G ] ∗ K )) the left horizontalmap in (10.9.1), and u = d − d : H ∗ ([ X × F/G ] , [pr / id G ] ∗ K ) → H ∗ ([ X × F × F/G ] , [pr / id G ] ∗ K )(resp. v = d − d : R ∗ G ( X × F, [pr / id G ] ∗ K ) → R ∗ G ( X × F × F, pr ∗ K )), the map deduced fromthe double map ( d , d ) in (10.9.1). As d and d are compatible with raising to the ℓ -th power,so is u (resp. v ). Let N = max { d + 2 e − , } . By Corollary 10.7 we have v ( y ) ℓ N = a ( x )for some x ∈ H ∗ ([ X × F/G ] , [pr / id G ] ∗ K ). By (10.9.1) we have a u ( x ) = v a ( x ) = 0. Let h be the least integer ≥ log ℓ (2 d + 2 f + 1). As above we have cd ℓ (( X × F × F ) /G ) ≤ d + 2 f , so byCorollary 10.7 we get u ( x ) ℓ h = 0, hence by (10.9.1) x ℓ h = u ( x ) for some x ∈ H ∗ ([ X/G ] , K ),and finally y ℓ N h = a ( x ). Remark 10.11. (a) If in case (a) of Corollary 10.10, we assume moreover that X is affine, then cd ℓ (( X × F ) /G ) ≤ d + e and cd ℓ (( X × F × F ) /G ) ≤ d + f by the affine Lefschetz theorem [50, XIV Corollaire3.2]. Thus in this case (i) holds for m = d + e + 1 and (ii) holds for N ≥ max { d + e − , } +log ℓ ( d + f + 1).(b) Let f : Y → X be a finite étale morphism of Artin stacks of constant degree d . As thecomposite H ∗ ( X , K ) f ∗ −→ H ∗ ( Y , f ∗ K ) tr f,K −−−→ H ∗ ( X , K ) is multiplication by d , f ∗ is injectiveif d is prime to ℓ . Thus, in this case, if Ker a Y ,f ∗ K is a nilpotent ideal, then Ker a X ,K isa nilpotent ideal with the same bound for the exponent. This applies in particular to themorphism [ X/H ] → [ X/G ], where
H < G is an open subgroup of index prime to ℓ . Proposition 10.12.
Theorem 8.3 (b) holds if X is a Deligne-Mumford stack of finite inertia.More precisely, if c = cd ℓ ( Y ) , where Y is the coarse moduli space of X , and if r (resp. s ) is themaximal number of elements of the inertia groups (resp. ℓ -Sylow subgroups of the inertia groups)of X , then (Ker a ( X , K )) ( c +1)(( s − +1) = 0 , and for K commutative and b ∈ R ∗ ( X , K ) , we have b ℓ N ∈ Im a ( X , K ) for N ≥ max { c − , } +max { r − r, } + ⌈ log ℓ (2( r − +1) ⌉ + ⌈ log ℓ (( s − +1) ⌉ .Here ⌈ x ⌉ for a real number x denotes the least integer ≥ x . roof. Consider the coarse moduli space morphism f : X → Y . For every geometric point t of Y ,there exists a finite radicial extension t ′ → t and a geometric point y ′ of X above t ′ such that( X t ′ ) red ≃ B Aut X ( y ′ ). Note that for any field E , a finite group G of order m can be embeddedinto GL m ( E ), given for example by the regular representation E [ G ] of G . Moreover, if m =2 or the characteristic of E is not 2, then G can be embedded into GL m − ( E ), because thesubrepresentation of E [ G ] generated by g − h , where g, h ∈ G , is faithful. Thus, by Remark 10.11,the map a X t ,K in Proposition 10.4 (c) satisfies (Ker a X t ,K ) ( s − +1 = 0, and, for K commutative, a X t ,K is a uniform F -surjection for all geometric points t → Y with bound for the exponent givenby max { r − r, } + ⌈ log ℓ (2( r − + 1) ⌉ , independent of t . Thus (Ker lim ←− t ∈ P Y a X t ,K ) ( s − +1 = 0,and Lemma 10.13 below implies that lim ←− t ∈ P Y a X t ,K is a uniform F -surjection, with bound for theexponent given by max { r − r, } + ⌈ log ℓ (2( r − + 1) ⌉ + ⌈ log ℓ (( s − + 1) ⌉ . Hence, by Lemma10.2 and Proposition 10.4, a X ,K has the stated properties. Lemma 10.13.
Let C be a category, and let u : R → S be a homomorphism of pseudo-rings in GrVec C . If u is a uniform F -injection (resp. uniform F -isomorphism) (Definition 6.10), then lim ←− C u is also a uniform F -injection (resp. uniform F -isomorphism). More precisely, if m ≥ isan integer such that for every object i of C and every a ∈ Ker u i , a m = 0 (resp. and if n ≥ is aninteger such that for every object i of C and every b ∈ S i , b ℓ n ∈ Im u i ), then for every x ∈ Ker lim ←− C u , x m = 0 (resp. for every y ∈ lim ←− C S and every integer N ≥ n + log ℓ ( m ) , y N ∈ Im lim ←− C u ).Proof. Let x = ( x i ) be an element in the kernel of lim ←− C u . Since x i is in Ker u i , x m = ( x mi ) = 0.Assume now that u is a uniform F -isomorphism with bounds for the exponents given by m and n , and let y = ( y i ) be an element of lim ←− C S . For every object i of C , take a i in R i such that u i ( a i ) = y ℓ n i . For every morphism α : i → j in C , the following diagram commutes R j u j / / R α (cid:15) (cid:15) S jS α (cid:15) (cid:15) R i u i / / S i . It follows that u i ( R α ( a j ) − a i ) = S α ( u j ( a j )) − u i ( a i ) = S α ( y ℓ n j ) − y ℓ n i = 0 . Let h be the least integer ≥ log ℓ ( m ). Then 0 = ( R α ( a j ) − a i ) ℓ h = R α ( a j ) ℓ h − a ℓ h i , so that w = ( a ℓ h i )is an element of lim ←− C R . By definition, u ( w ) = y ℓ n + h .In order to deal with the general case, we need the following lemma. Lemma 10.14.
Let u : R → S be a homomorphism of pseudo-rings in GrVec C endowed with asplitting (Definition 3.2). Then (Ker u ) R = 0 . In particular, (Ker u ) = 0 .Proof. Let a ∈ Ker u , b ∈ R . Since u ( a ) = 0, ab = u ( a ) b = 0. Proposition 10.15.
The first assertion of Theorem 8.3 (b) holds.Proof. If i : Y → X is a closed immersion, j : U → X is the complement, then the following diagramof graded rings commutes: H ∗ ( Y , Ri ! K ) / / a Y ,Ri ! K (cid:15) (cid:15) H ∗ ( X , K ) a X ,K (cid:15) (cid:15) / / H ∗ ( U , j ∗ K ) a U ,j ∗ K (cid:15) (cid:15) R ∗ ( Y , Ri ! K ) u / / R ∗ ( X , K ) / / R ∗ ( U , K ) . The first row is exact and u is the composition of the inverse of the isomorphism R ∗ ( X , i ∗ Ri ! K ) ∼ −→ R ∗ ( Y , Ri ! K ) and the map R ∗ ( X , i ∗ Ri ! K ) → R ∗ ( X , K ) induced by adjunction i ∗ Ri ! K → K . Thecomposition R ∗ ( Y , Ri ! K ) u −→ R ∗ ( X , K ) → R ∗ ( Y , i ∗ K )67s induced by Ri ! K → i ∗ K , hence has square-zero kernel by Lemma 10.14. Thus (Ker u ) = 0.It follows that if both a Y ,Ri ! K and a U ,j ∗ K have nilpotent kernels, then a X ,K has nilpotent kernel.Using this, we reduce by induction to the global quotient case. In this case, the assertion followsfrom Propositions 10.8 and 10.9.This finishes the proof of the structure theorem (Theorem 8.3 (b)). Lemma 10.16.
Let C be a category having finitely many isomorphism classes of objects. Let A be the category whose objects are the elementary abelian ℓ -groups and whose morphisms are themonomorphisms. Let F : C → A be a functor. Let F be the presheaf of F ℓ -algebras on A givenby F ( A ) = S( A ∨ ) . Let G be a presheaf of F ∗ F -modules on C . Assume that, for every object x of C , G ( x ) is a finitely generated F ( F ( x )) -module. Then R = lim ←− x ∈C F ( F ( x )) is a finitely generated F ℓ -algebra and S = lim ←− x ∈C G ( x ) is a finitely generated R -module.Proof. We may assume that C has finitely many objects. For any monomorphism u : A → B ofelementary abelian ℓ -groups, F ( u ) : F ( B ) → F ( A ) carries S( B ∨ ) GL( B ) into S( A ∨ ) GL( A ) . Thus A
7→ E ( A ) = S( A ∨ ) GL( A ) ⊂ F ( A ) defines a subpresheaf E of F ℓ -algebras of F . As GL( A ) is a finitegroup, by [48, V Corollaire 1.5] F ( A ) is finite over E ( A ) and E ( A ) is a finitely generated F ℓ -algebra.For given A and B , since GL( B ) acts transitively on the set of monomorphisms u : A → B , themap S( B ∨ ) GL( B ) → S( A ∨ ), restriction of F ( u ), does not depend on u . Thus E ( u ) only dependson A and B . Therefore, via the functor rk : A → N carrying A to its rank, E factorizes through apresheaf R on the totally ordered set N : we have a 2-commutative diagram C F / / f (cid:15) (cid:15) A E (cid:15) (cid:15) rk } } ④④④④④④④④ N R / / B op , where B denotes the category of F ℓ -algebras of finite type, and R ( n ) = S(( F nℓ ) ∨ ) GL n ( F ℓ ) , with, for m ≤ n , F mℓ included in F nℓ by any monomorphism. For a morphism u : A → B of A , F ( u ) : F ( B ) →F ( A ) is surjective, hence, as F ( B ) is finite over E ( B ), F ( A ) is finite over E ( B ), and E ( A ) ⊂ F ( A )is finite over E ( B ). By Lemma 10.19 below, for each x in C , E ( F ( x )) is finite over Q = lim ←− y ∈C E ( F ( y )) ≃ lim ←− y ∈C R ( f ( y )) . The rest of the proof is similar to the proof of the last assertion of loc. cit. As C has finitely manyobjects, there exists a finitely generated F ℓ -subalgebra Q of Q such that, for each x in C , E ( F ( x ))is integral, hence finite over Q . Note that R is a Q -submodule, a fortiori a Q -submodule, of Q x ∈C F ( F ( x )). For each x in C , F ( F ( x )) is finite over E ( F ( x )), hence finite over Q . It followsthat Q x ∈C F ( F ( x )) is finite over Q . As Q is a noetherian ring, R is finite over Q , hence a finitelygenerated F ℓ -algebra. Similarly, S is a finitely generated Q -module, hence a finitely generated R -module. Note that Q is also finite over Q , hence a finitely generated F ℓ -algebra, though we donot need this fact.The first step of the proof of Lemma 10.19 consists of simplifying the limit Q using cofinality.Among the functors f : • ( • • ) / / / / f : •• O O O O / / •• O O f : •• G G ✎✎ • W W ✴✴ / / •• O O f : •• W W ✴✴ • G G ✎✎ / / •• O O f , f , and f are cofinal, while f is not cofinal. It turns out that after making contractions oftypes f , f , and f , we obtain a rooted forest, of which the source of f is a prototype.For convenience we adopt the following order-theoretic definitions. We define a rooted forest to be a partially ordered set P such that P ≤ x = { y ∈ P | y ≤ x } is a finite chain for all x ∈ P .We define a rooted tree to be a nonempty connected rooted forest. Let P be a rooted tree. For x, y ∈ P , we say that y is a child of x if x < y and there exists no z ∈ P such that x < z < y . Bythe connectedness of P , m ( x ) = min P ≤ x is independent of x ∈ P , hence P has a least element r ,equal to m ( x ) for all x . We call r the root of P .68 emark 10.17. Although we do not need it, let us recall the comparison with graph-theoreticdefinitions. A graph-theoretic rooted tree T is a connected acyclic (undirected) graph with onevertex designated as the root [41, page 30]. For a graph-theoretic rooted tree T , we let V ( T )denote the set of vertices of T equipped with the tree-order, with x ≤ y if and only if the uniquepath from the root r to y passes through x . For any x ∈ V ( T ), V ( T ) ≤ x consists of vertices onthe path from r to x , so that V ( T ) ≤ x is a finite chain. Thus V ( T ) is a rooted tree. Conversely,for any rooted tree P , we construct a graph-theoretic rooted tree Γ( P ) as follows. Let G be thegraph whose set of vertices is P and such that two vertices x and y are adjacent if and only if y isa child of x or x is a child of y . Note that each x ≤ x ′ in P can be decomposed into a sequence x = x < x < · · · < x n = x ′ , n ≥
0, each x i +1 being a child of x i , which defines a path from x to x ′ in G . Thus the connectedness of P implies the connectedness of G . If G admits a cycle,then there exists y ∈ P that is a child of distinct elements x and x ′ of P , which contradicts theassumption that P ≤ y is a chain. Let r be the root of P . Then Γ( P ) = ( G, r ) is a graph-theoreticrooted tree. We have P = V (Γ( P )) and T = Γ( V ( T )).The next lemma is probably standard but we could not find an adequate reference. Lemma 10.18.
Let C be a category and let f : C → N be a functor. Let P be the set of fullsubcategories of C that are connected components of f − ( N ≥ n ) for some n ∈ N . Order P byinverse inclusion: for elements S and T of P , we write S ≤ T if S ⊃ T . Let ψ : C → P be thefunctor carrying an object x to the connected component ψ ( x ) of f − ( N ≥ f ( x ) ) containing x , andlet φ : P → N be the functor carrying S to min f ( S ) . Then:(a) f = φψ .(b) ψ : C → P is cofinal (Definition 6.1) and φ : P → N is strictly increasing.(c) P is a rooted forest. Moreover, if C has finitely many isomorphism classes of objects, then P is a finite set.Proof. (a) Let x be an object of C . As x ∈ ψ ( x ), φ ( ψ ( x )) = min f ( ψ ( x )) ≤ f ( x ). Conversely, as ψ ( x ) ⊂ f − ( N ≥ f ( x ) ), f ( ψ ( x )) ⊂ N ≥ f ( x ) , so that φ ( ψ ( x )) ≥ f ( x ). Thus φ ( ψ ( x )) = f ( x ).(b) Let S ∈ P . Note that S is a connected component of f − ( N ≥ φ ( S ) ). By definition, ( S ↓ ψ )is the category of pairs ( x, S ≤ ψ ( x )). Note that S ⊃ ψ ( x ) implies that x is in S . Conversely, for x in S , S is a connected component of f − ( N ≥ n ) for n ≤ f ( x ), hence S ⊃ ψ ( x ). Thus ( S ↓ ψ ) canbe identified with S , hence is connected. This shows that ψ is cofinal. Now let S < T be elementsof P . We have φ ( S ) ≤ φ ( T ). If φ ( S ) = φ ( T ) = n , then S and T are both connected componentsof f − ( N ≥ n ), which contradicts with the assumption S ) T . Thus φ ( S ) < φ ( T ).(c) Let S ∈ P . Let T, T ′ ∈ P ≤ S . Then T (resp. T ′ ) is a connected components of f − ( N ≥ n )(resp. f − ( N ≥ n ′ )), and T and T ′ both contain S . Thus T ⊃ T ′ if n ≤ n ′ and T ⊂ T ′ if n ≥ n ′ .Therefore, P ≤ S is a chain. As φ is strictly increasing, φ induces an injection P ≤ S → N ≤ φ ( S ) , hence P ≤ S is a finite set. Therefore, P is a rooted forest. Note that for S ∈ P and x in S , every object y of C isomorphic to x is also in S . Thus, if C has finitely many isomorphism classes of objects, then P is a finite set. Lemma 10.19.
Let C be a category having finitely many isomorphism classes of objects and let f : C → N be a functor. Let R be a presheaf of commutative rings on N such that, for each m ≤ n , R ( m ) is finite over R ( n ) . Let Q = lim ←− x ∈C R ( f ( x )) . Then:(a) For each object x of C , R ( f ( x )) is finite over Q .(b) For each connected component S of C and each r in S satisfying f ( r ) = min f ( S ) , we have Im( Q → R ( f ( r ))) = Im( R (max f ( S )) → R ( f ( r ))) . Proof.
By Lemma 10.18, we may assume that C is a finite rooted tree with root r . We prove thiscase by induction on C . Let B ⊂ C be the set of children of r . For each c ∈ B , C ≥ c is a rooted treewith root c and Q is the fiber product over R ( f ( r )) of the rings Q c = lim ←− x ∈C ≥ c R ( f ( x )) for c ∈ B .If B is empty, then C = { r } and the assertions are trivial. If B = { c } , then Q ≃ Q c and it sufficesto apply the induction hypothesis to Q c . Assume B >
1. Let n = max f ( C ), n c = max f ( C ≥ c ),and let c ∈ B be such that n c = min c ∈ B n c . The complement C ′ of C ≥ c in C is a rooted treewith root r , and Q is the fiber product over R ( f ( r )) of the rings Q c and Q ′ = lim ←− x ∈C ′ R ( f ( x )).69y the induction hypothesis, A = Im( Q c → R ( f ( r ))) = Im( R ( n c ) → R ( f ( r ))) and Im( Q ′ →R ( f ( r ))) = Im( R ( n ) → R ( f ( r ))), so that we have a cartesian square of commutative rings Q α ′ / / β ′ (cid:15) (cid:15) Q ′ β (cid:15) (cid:15) Q c α / / A. As α is surjective, we have Ker( α ′ ) ≃ Ker( α ) and α ′ is surjective (cf. [17, Lemme 1.3]), whichimplies (b). Moreover, as β is finite, β ′ is finite. Indeed, if A = P i a i β ( Q ′ ), then for liftings a ′ i of a i , Q c = P i a ′ i β ′ ( Q ). The assertion (a) then follows from the induction hypothesis applied to Q c and Q ′ . Proof of Theorem 8.3 (a).
Let ( j i : X i → X ) i be a finite stratification of X by locally closed sub-stacks. The system of functors ( C X i → C X ) i is essentially surjective. Thus the map R ∗ ( X , K ) → Y i R ∗ ( X i , j ∗ i K )is an injection. Thus, for the first assertion of Theorem 8.3 (a), we may assume that X is a globalquotient stack, in which case the assertion follows from Theorem 6.17 (a) and Proposition 8.7.Let H q ( K • ) denote the presheaf on C X whose value at x : S → X is H q ( S , K x ), where K x = x ∗ K , so that R q ( X , K ) = lim ←− C X H q ( K • ). Let N be the set of morphisms f in C X such that ( H ∗ ( F ℓ • ))( f ) and ( H ∗ ( K • ))( f ) are isomorphisms. By Lemma 7.3, lim ←− C X H q ( K • ) ≃ lim ←− N − C X H q ( K • ) and similarly for H ∗ ( F ℓ • ). We claim that N − C X has finitely many isomor-phism classes of objects. Then lim ←− N − C X commutes with direct sums, and, by Lemma 10.16, R ∗ ( X , F ℓ ) and R ∗ ( X , K ) are finitely generated R -modules for a finitely-generated F ℓ -algebra R ,hence the second assertion of Theorem 8.3 (a). Using again the fact that the system of functors( C X i → C X ) i is essentially surjective, we may assume in the above claim that X = [ X/G ] is aglobal quotient. Consider the diagram (8.5.2). Note that the functor A G ( k ) ♮ → E G ( π ) inducesa bijection between the sets of isomorphism classes of objects, and E G ( π ) is essentially finite byLemma 6.20 (a), thus A G ( k ) ♮ has finitely many isomorphism classes of objects. Moreover, as E isessentially surjective, it suffices to show that, for every object ( A, A ′ , g ) of A G ( k ) ♮ , the category M − P X A ′ has finitely many isomorphism classes of objects. Here M is the set of morphisms f in P X A ′ such that ( E ∗ ( A,A ′ ,g ) H ∗ ( K • ))( f ) is an isomorphism. Let ( X i ) be a finite stratification of X A ′ into locally closed subschemes such that K | X i has locally constant cohomology sheaves. For agiven i , all objects in the image of P X i → M − P X A ′ are isomorphic. Moreover, the system of func-tors ( P X i → P X A ′ ) i is essentially surjective. Therefore, M − P X A ′ has finitely many isomorphismclasses of objects.
11 Stratification of the spectrum
In this section we fix an algebraically closed field k and a prime number ℓ invertible in k . Construction 11.1.
Let X be a separated algebraic space of finite type over k , and let G be analgebraic group over k acting on X . Define(11.1.1) ( G, X ) := Spec H ε ∗ ([ X/G ]) red , where ε = 1 if ℓ = 2, and ε = 2 otherwise. In particular, for an elementary abelian ℓ -group A , A := ( A, Spec k ) = Spec( H ε ∗ A ) red is a standard affine space of dimension equal to the rank of A . The map ( A, C ) ∗ (6.9.2) induces amorphism of schemes(11.1.2) ( A, C ) ∗ : A → ( G, X ) , a ( G, X ) (6.9.3) induces a morphism of schemes(11.1.3) Y := lim −→ ( A,C ) ∈A ♭ ( G,X ) A → ( G, X ) . It follows from Theorem 6.11 that (11.1.3) is a universal homeomorphism.By Corollary 4.8, (
A, C ) ∗ is finite. Moreover, A ♭ ( G,X ) is essentially finite by Lemma 6.5. Itfollows that Y ≃ Spec(lim ←− ( A,C ) ∈A ♭ ( G,X ) ( H ε ∗ A ) red ) is finite over ( G, X ) and is a colimit of A in thecategory of locally ringed spaces and in particular a colimit of A in the category of schemes. Thisremark gives another proof of the second assertion of Corollary 6.13, as promised. Moreover,the F ℓ -algebras H ε ∗ ([ X/G ]) red and lim ←− ( A,C ) ∈A ♭ ( G,X ) ( H ε ∗ A ) red are equipped with Steenrod opera-tions (see Construction 11.6 below), compatible with the ring homomorphism H ε ∗ ([ X/G ]) red → lim ←− ( A,C ) ∈A ♭ ( G,X ) ( H ε ∗ A ) red .The structure of Y is described more precisely by the following stratification theorem , similarto [37, Theorems 10.2, 12.1]. Theorem 11.2.