Odds and ends on equivariant cohomology and traces Weizhe Zheng Columbia University International Congress of Chinese Mathematicians Tsinghua University, Beijing December 18, 2010 Joint work with Luc Illusie. Weizhe Zheng Equivariant cohomology and traces ICCM 2010 1 / 26
Introduction Introduction Let k be an algebraically closed field of characteristic p ≥ 0, X be a separated scheme of finite type over k , G be a finite group acting on X . For any prime number ℓ � = p , H i ( X , Q ℓ ) is a finite-dimensional ℓ -adic representation of G . For g ∈ G , ( − 1) i Tr( g , H i ( X , Q ℓ )) ∈ Z ℓ . � t ℓ ( g ) := i Weizhe Zheng Equivariant cohomology and traces ICCM 2010 2 / 26
Introduction Introduction Let k be an algebraically closed field of characteristic p ≥ 0, X be a separated scheme of finite type over k , G be a finite group acting on X . For any prime number ℓ � = p , H i ( X , Q ℓ ) is a finite-dimensional ℓ -adic representation of G . For g ∈ G , ( − 1) i Tr( g , H i ( X , Q ℓ )) ∈ Z ℓ . � t ℓ ( g ) := i Problem Is t ℓ ( g ) in Z and independent of ℓ ? Problem i ( − 1) i [ H i ( X , Q ℓ )] Describe the virtual representation χ ( X , G , Q ℓ ) := � of G under suitable assumptions on the action of G. Weizhe Zheng Equivariant cohomology and traces ICCM 2010 2 / 26
Plan of the talk Plan of the talk Generalization of Laumon’s theorem on Euler characteristics 1 Tameness at infinity 2 Mod ℓ equivariant cohomology algebra 3 Weizhe Zheng Equivariant cohomology and traces ICCM 2010 3 / 26
Generalization of Laumon’s theorem on Euler characteristics Plan of the talk Generalization of Laumon’s theorem on Euler characteristics 1 Tameness at infinity 2 Mod ℓ equivariant cohomology algebra 3 Weizhe Zheng Equivariant cohomology and traces ICCM 2010 4 / 26
Generalization of Laumon’s theorem on Euler characteristics Compactly supported cohomology Compactly supported cohomology H i c ( X , Q ℓ ) is also a finite-dimensional ℓ -adic representation of G . For g ∈ G , ( − 1) i Tr( g , H i � c ( X , Q ℓ )) ∈ Z ℓ . t c , X ,ℓ ( g ) := i Additivity: If Z ⊂ X is a G -stable closed subscheme, then t c , X ,ℓ ( g ) = t c , Z ,ℓ ( g ) + t c , X − Z ,ℓ ( g ) . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 5 / 26
Generalization of Laumon’s theorem on Euler characteristics Compactly supported cohomology Compactly supported cohomology H i c ( X , Q ℓ ) is also a finite-dimensional ℓ -adic representation of G . For g ∈ G , ( − 1) i Tr( g , H i � c ( X , Q ℓ )) ∈ Z ℓ . t c , X ,ℓ ( g ) := i Additivity: If Z ⊂ X is a G -stable closed subscheme, then t c , X ,ℓ ( g ) = t c , Z ,ℓ ( g ) + t c , X − Z ,ℓ ( g ) . Theorem (Deligne-Lusztig 1976) t c ,ℓ ( g ) is in Z and independent of ℓ . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 5 / 26
Generalization of Laumon’s theorem on Euler characteristics Compactly supported cohomology Theorem (1) t ℓ ( g ) = t c ,ℓ ( g ) . Corollary t ℓ ( g ) is in Z and independent of ℓ . If g = 1, ( ?? ) becomes χ ( X , Q ℓ ) = χ c ( X , Q ℓ ) , which follows from Laumon’s theorem. Weizhe Zheng Equivariant cohomology and traces ICCM 2010 6 / 26
Generalization of Laumon’s theorem on Euler characteristics Laumon’s theorem Laumon’s theorem Let k be an arbitrary field of characteristic p . For X separated of finite type over k , let D b c ( X , Q ℓ ) be the category of (bounded) ℓ -adic complexes, K ( X , Q ℓ ) be the corresponding Grothendieck ring, K ∼ ( X , Q ℓ ) be the quotient of K ( X , Q ℓ ) by the ideal generated by [ Q ℓ (1)] − [ Q ℓ ]. For any morphism f : X → Y , the exact functors Rf ∗ , Rf ! : D b c ( X , Q ℓ ) → D b c ( Y , Q ℓ ) induce group homomorphisms f ∗ , f ! : K ( X , Q ℓ ) → K ( Y , Q ℓ ) , f ∼ ∗ , f ∼ ! : K ∼ ( X , Q ℓ ) → K ∼ ( Y , Q ℓ ) . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 7 / 26
Generalization of Laumon’s theorem on Euler characteristics Laumon’s theorem Laumon’s theorem Let k be an arbitrary field of characteristic p . For X separated of finite type over k , let D b c ( X , Q ℓ ) be the category of (bounded) ℓ -adic complexes, K ( X , Q ℓ ) be the corresponding Grothendieck ring, K ∼ ( X , Q ℓ ) be the quotient of K ( X , Q ℓ ) by the ideal generated by [ Q ℓ (1)] − [ Q ℓ ]. For any morphism f : X → Y , the exact functors Rf ∗ , Rf ! : D b c ( X , Q ℓ ) → D b c ( Y , Q ℓ ) induce group homomorphisms f ∗ , f ! : K ( X , Q ℓ ) → K ( Y , Q ℓ ) , f ∼ ∗ , f ∼ ! : K ∼ ( X , Q ℓ ) → K ∼ ( Y , Q ℓ ) . Theorem (Laumon 1981) f ∼ ∗ = f ∼ ! . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 7 / 26
Generalization of Laumon’s theorem on Euler characteristics Equivariant complexes Equivariant complexes For X separated of finite type over k and G finite acting on X , let D b c ( X , G , Q ℓ ) be the category of (bounded) G -equivariant ℓ -adic complexes, K ( X , G , Q ℓ ) be the corresponding Grothendieck ring, K ∼ ( X , G , Q ℓ ) be the quotient of K ( X , G , Q ℓ ) by the ideal generated by [ Q ℓ (1)] − [ Q ℓ ]. Let ( f , u ): ( X , G ) → ( Y , H ), where u : G → H is a homomorphism and f : X → Y is a u -equivariant morphism. The exact functors R ( f , u ) ∗ , R ( f , u ) ! : D b c ( X , G , Q ℓ ) → D b c ( Y , H , Q ℓ ) induce group homomorphisms ( f , u ) ∗ , ( f , u ) ! : K ( X , G , Q ℓ ) → K ( Y , H , Q ℓ ) , ( f , u ) ∼ ∗ , ( f , u ) ∼ ! : K ∼ ( X , G , Q ℓ ) → K ∼ ( Y , H , Q ℓ ) . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 8 / 26
Generalization of Laumon’s theorem on Euler characteristics Equivariant complexes Theorem ( f , u ) ∼ ∗ = ( f , u ) ∼ ! . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 9 / 26
Generalization of Laumon’s theorem on Euler characteristics Equivariant complexes Theorem ( f , u ) ∼ ∗ = ( f , u ) ∼ ! . One key step of the proof is the following. Proposition Let S be the spectrum of a henselian discrete valuation ring, with closed point s = Spec( k ) and generic point η . Then for any L ∈ D b c ( X × s η, G , Q ℓ ) , the class in K ∼ ( X , G , Q ℓ ) of R Γ( I , L ) ∈ D b c ( X , G , Q ℓ ) is zero, where I is the inertia subgroup of the Galois group of η . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 9 / 26
Generalization of Laumon’s theorem on Euler characteristics Complexes on Deligne-Mumford stacks Complexes on Deligne-Mumford stacks Let S be a regular (Noetherian) scheme of dimension ≤ 1, ℓ be a prime invertible on S . One can define, for every Deligne-Mumford stack X of finite type over S , a category D b c ( X , Q ℓ ) of ℓ -adic complexes on X , and, for every morphism f : X → Y , exact functors Rf ∗ , Rf ! : D b c ( X , Q ℓ ) → D b c ( Y , Q ℓ ) , f ∗ , Rf ! : D b c ( Y , Q ℓ ) → D b c ( X , Q ℓ ) . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 10 / 26
Generalization of Laumon’s theorem on Euler characteristics Complexes on Deligne-Mumford stacks Complexes on Deligne-Mumford stacks Let S be a regular (Noetherian) scheme of dimension ≤ 1, ℓ be a prime invertible on S . One can define, for every Deligne-Mumford stack X of finite type over S , a category D b c ( X , Q ℓ ) of ℓ -adic complexes on X , and, for every morphism f : X → Y , exact functors Rf ∗ , Rf ! : D b c ( X , Q ℓ ) → D b c ( Y , Q ℓ ) , f ∗ , Rf ! : D b c ( Y , Q ℓ ) → D b c ( X , Q ℓ ) . Under an additional condition of finiteness of cohomological dimension, Laszlo-Olsson 2008 defined an ℓ -adic formalism for unbounded complexes on Artin stacks. Weizhe Zheng Equivariant cohomology and traces ICCM 2010 10 / 26
Generalization of Laumon’s theorem on Euler characteristics Complexes on Deligne-Mumford stacks For a Deligne-Mumford stack X of finite type over S , let K ( X , Q ℓ ) be the Grothendieck ring of D b c ( X , Q ℓ ), K ∼ ( X , Q ℓ ) be the quotient by the ideal generated by [ Q ℓ (1)] − [ Q ℓ ]. For f : X → Y , Rf ∗ and Rf ! induce group homomorphisms f ∼ ∗ , f ∼ ! : K ∼ ( X , Q ℓ ) → K ∼ ( Y , Q ℓ ) , f ∗ and Rf ! induce group homomorphisms f ∗∼ , f ! ∼ : K ∼ ( Y , Q ℓ ) → K ∼ ( X , Q ℓ ) . f ∗∼ is a ring homomorphism. Weizhe Zheng Equivariant cohomology and traces ICCM 2010 11 / 26
Generalization of Laumon’s theorem on Euler characteristics Complexes on Deligne-Mumford stacks For a Deligne-Mumford stack X of finite type over S , let K ( X , Q ℓ ) be the Grothendieck ring of D b c ( X , Q ℓ ), K ∼ ( X , Q ℓ ) be the quotient by the ideal generated by [ Q ℓ (1)] − [ Q ℓ ]. For f : X → Y , Rf ∗ and Rf ! induce group homomorphisms f ∼ ∗ , f ∼ ! : K ∼ ( X , Q ℓ ) → K ∼ ( Y , Q ℓ ) , f ∗ and Rf ! induce group homomorphisms f ∗∼ , f ! ∼ : K ∼ ( Y , Q ℓ ) → K ∼ ( X , Q ℓ ) . f ∗∼ is a ring homomorphism. Theorem f ∼ ∗ = f ∼ ! . Corollary f ∗∼ = f ! ∼ . Weizhe Zheng Equivariant cohomology and traces ICCM 2010 11 / 26
Tameness at infinity Plan of the talk Generalization of Laumon’s theorem on Euler characteristics 1 Tameness at infinity 2 Mod ℓ equivariant cohomology algebra 3 Weizhe Zheng Equivariant cohomology and traces ICCM 2010 12 / 26
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