Deligne-Conjecture2 - FUJIWARA’S THEOREM FOR EQUIVARIANT...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ) ) ....
View Full Document

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.

Page1 / 82

Deligne-Conjecture2 - FUJIWARA’S THEOREM FOR EQUIVARIANT...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online