{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Deligne-Conjecture2

Deligne-Conjecture2 - FUJIWARAS THEOREM FOR EQUIVARIANT...

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

View Full Document Right Arrow Icon
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 0 denotes a scheme (or stack) over F q and X denotes the fiber product X 0 × Spec( F q ) Spec( k ). Let X 0 be a separated finite type F q -scheme. A correspondence on X 0 is a diagram of separated finite type F q -schemes C 0 c 1 c 2 X 0 X 0 , or equivalently a morphism c = ( c 1 , c 2 ) : C 0 X 0 × X 0 . For n 0 we write c ( n ) = ( c ( n ) 1 , c 2 ) : C ( n ) 0 X 0 × X 0 for the correspondence C 0 c ( n ) 1 c 1 c 2 X 0 F n X 0 X 0 , X 0 where F X 0 : X 0 X 0 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
Image of page 1

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

View Full Document Right Arrow Icon
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 0 , independent of ( F , ϕ, u ) , such that for any integer n n 0 all the fixed points of c ( n ) are isolated, and (1.1.1) tr( c ( n ) | R Γ c ( X, F )) = x Fix( C ( n ) ( k )) tr( u ( n ) x |F c 2 ( x ) ) . Remark 1.2. Note that the right side of 1.1.1 is a finite sum. With the recent work on cohomology with compact supports for Artin stacks [13, 14], it is natural to ask for a generalization of 1.1 to Artin stacks. In this paper we propose a conjectural generalization for arbitrary stacks, and we prove this conjecture in a number of cases (in particular for equivariant correspondences). Fujiwara’s theorem is most naturally viewed in two parts. The first part is a geometric statement that the fixed points of c ( n ) are isolated and that the sum of the “naive local terms” x Fix( C ( n ) ( k )) tr( u ( n ) x |F c 2 ( x ) ) is equal to the sum of the “true local terms” as defined in [3, III § 4]. The second part is a reduction to the Lefschetz trace formula [3, III.4.7], which holds
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern