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Unformatted text preview: FUJIWARA’S THEOREM FOR EQUIVARIANT CORRESPONDENCES MARTIN OLSSON 1. Statements of results The subject of this paper is a generalization to stacks of Fujiwara’s theorem [10, 5.4.5] (formerly known as Deligne’s conjecture) on the traces of a correspondence acting on the compactly supported cohomology of a variety over a finite field. Before discussing the stack-theoretic version, let us begin by reviewing Fujiwara’s theorem. Let q be a power of a prime p , and let k = F q be an algebraic closure of F q . For objects over F q we use a subscript 0, and unadorned letters denote the base change to k . For example, X denotes a scheme (or stack) over F q and X denotes the fiber product X × Spec( F q ) Spec( k ). Let X be a separated finite type F q-scheme. A correspondence on X is a diagram of separated finite type F q-schemes C c 1 ~ | | | | | | | | c 2 ! C C C C C C C C X X , or equivalently a morphism c = ( c 1 ,c 2 ) : C → X × X . For n ≥ 0 we write c ( n ) = ( c ( n ) 1 ,c 2 ) : C ( n ) → X × X for the correspondence C c ( n ) 1 c 1 ~ | | | | | | | | c 2 ! C C C C C C C C X F n X ~ | | | | | | | | X , X where F X : X → X denotes the q-th power Frobenius morphism. We write Fix( C ) (or sometimes Fix( c ) if we want to emphasize the reference to the mor- phism c ) for the fiber product of the diagram (over k ) C c X Δ / X × Spec( k ) X. 1 2 MARTIN OLSSON If F ∈ D b c ( X, Q ` ) we define a C-structure on F to be a map u : c 2! c * 1 F → F in D b c ( X, Q ` ) (or equivalently a map c * 1 F → c ! 2 F in D b c ( C, Q ` )). A Weil complex on X is a pair ( F ,ϕ ), where F ∈ D b c ( X, Q ` ) and ϕ : F * X F → F is an isomorphism. If ( F ,ϕ,u ) is a Weil complex with C-structure and n ≥ 0, then ( F ,ϕ ) has a C ( n )-structure given by the map u ( n ) : c 2! c ( n ) * 1 F = c 2! c * 1 F n * X F ϕ n / c 2! c * 1 F u / F . Assume now that c 1 is proper and that c 2 is quasi-finite. For a fixed point x ∈ Fix( C )( k ) we get for any Weil complex with C-structure ( F ,ϕ,u ) an endomorphism u x : F c 2 ( x ) → F c 2 ( x ) defined as follows. Since c 2 : C → X is quasi-finite, we have ( c 2! c * 1 F ) c 2 ( x ) = ⊕ y F c 1 ( y ) , where the sum is taken over the set of points y ∈ C ( k ) with c 2 ( y ) = c 2 ( x ). The map u x is defined to be the composite F c 2 ( x ) = F c 1 ( x ) x / ⊕ y F c 1 ( y ) ( c 2! c * 1 F ) c 2 ( x ) u / F c 2 ( x ) . Deligne’s conjecture, proven by Fujiwara, is then the following: Theorem 1.1 (Fujiwara [10, 5.4.5]) . There exists an integer n , independent of ( F ,ϕ,u ) , such that for any integer n ≥ n all the fixed points of c ( n ) are isolated, and (1.1.1) tr( c ( n ) | R Γ c ( X, F )) = X x ∈ Fix( C ( n ) ( k )) tr( u ( n ) x |F c 2 ( x ) ) ....
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This note was uploaded on 02/05/2011 for the course MATH 224b taught by Professor Grunbaum,f during the Spring '08 term at University of California, Berkeley.
- Spring '08