Mod ℓ cohomology algebras of quotient stacks Analogues of Quillen’s theory Weizhe Zheng Columbia University and Chinese Academy of Sciences Pan-Asian Number Theory Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing August 23, 2011 Joint work with Luc Illusie. Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 1 / 37
Introduction Plan of the talk Introduction 1 Cohomology of Artin stacks; finiteness 2 Structure theorems 3 Equivariant version with constant coefficients Equivariant version with general coefficients Stacky version A localization theorem 4 Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 2 / 37
Introduction Introduction Fix a prime number ℓ . F ℓ := Z /ℓ Z . Let G be a compact Lie group, BG be a classifying space of G . Consider the graded F ℓ -algebra H ∗ G ( F ℓ ) := H ∗ ( BG , F ℓ ) , satisfying a ∪ b = ( − 1) ij b ∪ a G ( F ℓ ), b ∈ H j for a ∈ H i G ( F ℓ ). The F ℓ -algebra H ǫ ∗ G ( F ℓ ) is commutative, where � 1 ℓ = 2 , ǫ = 2 ℓ > 2 . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 3 / 37
Introduction Definition An elementary abelian ℓ -group is a finite dimensional F ℓ -vector space. The rank of the group is the dimension of the vector space. Fact Let A ≃ ( Z /ℓ Z ) r . � F ℓ [ x 1 , . . . , x r ] ℓ = 2 , H ∗ A ( F ℓ ) = ∧ ( F ℓ x 1 ⊕ · · · ⊕ F ℓ x r ) ⊗ F ℓ [ y 1 , . . . , y r ] ℓ > 2 , where x 1 , . . . , x r form a basis of H 1 = Hom( A , F ℓ ) , y 1 , . . . , y r ∈ H 2 . In particular, Spec( H ǫ ∗ A ( F ℓ )) is homeomorphic to A r F ℓ . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 4 / 37
Introduction Quillen’s structure theorem Quillen’s structure theorem Let A be the category of elementary abelian ℓ -subgroups of G . A morphism A → A ′ in A is an element g ∈ G such that g − 1 Ag ⊂ A ′ . Theorem (Quillen) The homomorphism H ∗ H ∗ G ( F ℓ ) → lim A ( F ℓ ) ← − A ∈A is a uniform F-isomorphism. A homomorphism of F ℓ -algebras is called a uniform F -isomorphism if F N = 0 on the kernel and cokernel for N large enough. Here F : a �→ a ℓ . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 5 / 37
Introduction Quillen’s structure theorem Quillen’s structure theorem Let A be the category of elementary abelian ℓ -subgroups of G . A morphism A → A ′ in A is an element g ∈ G such that g − 1 Ag ⊂ A ′ . Theorem (Quillen) The homomorphism H ∗ H ∗ G ( F ℓ ) → lim A ( F ℓ ) ← − A ∈A is a uniform F-isomorphism. A homomorphism of F ℓ -algebras is called a uniform F -isomorphism if F N = 0 on the kernel and cokernel for N large enough. Here F : a �→ a ℓ . Corollary The Krull dimension of H ǫ ∗ G ( F ℓ ) is equal to the maximum rank of the elementary abelian ℓ -subgroups of G. This was conjectured by Atiyah and Swan. Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 5 / 37
Introduction Quillen’s structure theorem More generally, Quillen considered the equivariant cohomology algebra H ∗ G ( X , F ℓ ), where X is a topological space acted on by G . Theorem (Quillen) Assume X is paracompact and of finite ℓ -cohomological dimension. Then the homomorphism H ∗ H ∗ G ( X , F ℓ ) → lim A ( F ℓ ) ← − ( A , C ) is a uniform F-isomorphism. Here the limit is taken over pairs ( A , C ) , where A is an elementary abelian ℓ -subgroup of G, C is a connected component of the fixed point set X A . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 6 / 37
Introduction Algebraic setting Algebraic setting Fix an algebraically closed base field k of characteristic � = ℓ . Structure theorem for H ∗ ([ X / G ] , F ℓ ), where X is a scheme over k , G is an algebraic group over k acting on X , [ X / G ] is the quotient stack. Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 7 / 37
Introduction Algebraic setting Algebraic setting Fix an algebraically closed base field k of characteristic � = ℓ . Structure theorem for H ∗ ([ X / G ] , F ℓ ), where X is a scheme over k , G is an algebraic group over k acting on X , [ X / G ] is the quotient stack. Stacky interpretation of H ∗ ( M , F ℓ ), where M is a moduli stack over k . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 7 / 37
Introduction Algebraic setting Algebraic setting Fix an algebraically closed base field k of characteristic � = ℓ . Structure theorem for H ∗ ([ X / G ] , F ℓ ), where X is a scheme over k , G is an algebraic group over k acting on X , [ X / G ] is the quotient stack. Stacky interpretation of H ∗ ( M , F ℓ ), where M is a moduli stack over k . H ∗ ( M , R ∗ f ∗ F ℓ ), where f : T → M is a universal family. Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 7 / 37
Plan of the talk Plan of the talk Introduction 1 Cohomology of Artin stacks; finiteness 2 Structure theorems 3 Equivariant version with constant coefficients Equivariant version with general coefficients Stacky version A localization theorem 4 Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 8 / 37
Cohomology of Artin stacks Plan of the talk Introduction 1 Cohomology of Artin stacks; finiteness 2 Structure theorems 3 Equivariant version with constant coefficients Equivariant version with general coefficients Stacky version A localization theorem 4 Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 9 / 37
Cohomology of Artin stacks Cartesian sheaves Cartesian sheaves Let X be an Artin stack. Mod Cart ( X , F ℓ ) := lim − Mod( X ´ et , F ℓ ) , ← where the limit is taken over smooth morphisms X → X , where X is a scheme. If X 0 → X is a smooth presentation (i.e. a smooth surjection such that X 0 is a scheme), Mod Cart ( X , F ℓ ) ≃ lim Mod(( X n ) ´ et , F ℓ ) , ← − n where X • = cosk 0 ( X 0 / X ) ( X n is the fiber product of n + 1 copies of X 0 above X ). Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 10 / 37
Cohomology of Artin stacks Cartesian sheaves Cartesian sheaves Let X be an Artin stack. Mod Cart ( X , F ℓ ) := lim − Mod( X ´ et , F ℓ ) , ← where the limit is taken over smooth morphisms X → X , where X is a scheme. If X 0 → X is a smooth presentation (i.e. a smooth surjection such that X 0 is a scheme), Mod Cart ( X , F ℓ ) ≃ lim Mod(( X n ) ´ et , F ℓ ) , ← − n where X • = cosk 0 ( X 0 / X ) ( X n is the fiber product of n + 1 copies of X 0 above X ). Example If X is a Deligne-Mumford stack, Mod Cart ( X , F ℓ ) ≃ Mod( X ´ et , F ℓ ). Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 10 / 37
Cohomology of Artin stacks Cartesian sheaves Example Let X be a scheme over k , G be an algebraic group over k acting on X . The quotient stack [ X / G ] is an Artin stack and Mod Cart ([ X / G ] , F ℓ ) is the category of G -equivariant F ℓ -sheaves on X ´ et . Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 11 / 37
Cohomology of Artin stacks Cartesian sheaves Example Let X be a scheme over k , G be an algebraic group over k acting on X . The quotient stack [ X / G ] is an Artin stack and Mod Cart ([ X / G ] , F ℓ ) is the category of G -equivariant F ℓ -sheaves on X ´ et . Example BG = [Spec( k ) / G ]. Mod Cart ( BG , F ℓ ) is the category of F ℓ -representations of G . In particular, Mod Cart ( BG , F ℓ ) ≃ Mod Cart ( B π 0 ( G ) , F ℓ ) . Mod Cart ( X , F ℓ ) does not determine H ∗ ( X , F ℓ ). Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 11 / 37
Cohomology of Artin stacks Derived category of Cartesian sheaves Derived category of Cartesian sheaves Two approaches: 1 (Laumon-Moret-Bailly) Consider the site whose objects are smooth morphisms X → X where X is a scheme and whose covering families are smooth surjective families. It defines a topos X sm . Mod Cart ( X , F ℓ ) is a full subcategory of Mod( X sm , F ℓ ). Define D Cart ( X , F ℓ ) to be the triangulated full subcategory of D ( X sm , F ℓ ) consisting of complexes with cohomology sheaves in Mod Cart ( X , F ℓ ). Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 12 / 37
Cohomology of Artin stacks Derived category of Cartesian sheaves Derived category of Cartesian sheaves Two approaches: 1 (Laumon-Moret-Bailly) Consider the site whose objects are smooth morphisms X → X where X is a scheme and whose covering families are smooth surjective families. It defines a topos X sm . Mod Cart ( X , F ℓ ) is a full subcategory of Mod( X sm , F ℓ ). Define D Cart ( X , F ℓ ) to be the triangulated full subcategory of D ( X sm , F ℓ ) consisting of complexes with cohomology sheaves in Mod Cart ( X , F ℓ ). (Behrend, Gabber) X sm is not functorial. For a morphism f : X → Y of Artin stacks, f ∗ : Y sm → X sm is not left exact in general. (Olsson, Laszlo-Olsson) Define f ∗ : D Cart ( Y , F ℓ ) → D Cart ( X , F ℓ ) using smooth presentations and cohomological descent. Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 12 / 37
Cohomology of Artin stacks Derived category of Cartesian sheaves 2 (Liu-Zheng in progress) D Cart ( X , F ℓ ) := lim D (( X n ) ´ et , F ℓ ) ← − n is a presentable stable ∞ -category, independent (up to equivalences) of the choice of the smooth presentation X 0 → X . Here D (( X n ) ´ et , F ℓ ) is the derived ∞ -category of Mod(( X n ) ´ et , F ℓ ) defined by Lurie. Advantages: base change in derived categories (instead of on the level of sheaves); fewer finiteness assumptions Weizhe Zheng Mod ℓ cohomology algebras PANT 2011 13 / 37
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