Introduction

Motivation. The present paper grew out of the desire to prove the fol-

lowing theorem, conjectured in [3]

T H E O R E M

0.1 (Lefschetz trace formula for stacks). Let X be a smooth

algebraic stack over the finite field ¥q. Then

qdiraXt^q\H*(X,Qi) = # £ ( F

g

) . (1)

Here &q is the (arithmetic) Frobenius acting on the l-adic cohomology of X.

For example, if X = £ G

m

, the classifying stack of the multiplicative

group, then dimI?G

m

= —1 (since dividing the 'point' SpecF

g

of dimension

0 by the 1-dimensional group G

m

gives a quotient of dimension —1), and

tr$

g

|i? 2 z (jBG

m

,Q^) = i , (since the cohomology of BGm is the cohomology

of the infinite-dimensional projective space). Thus the left hand side of (1)

is

«hqi q~1'

On the right hand side we get the number of Fg-rational points of the

stack X. For the case of BGm this is the number of principal Gm-bundles over

SpecF

g

(up to isomorphism), or the number of line bundles over SpecF

g

.

Since all line bundles over SpecFg are trivial, there is only one isomorphism

class in BGm(¥q). But the number of automorphisms of the trivial line

bundle is #Gm(F?) = # F * = q — 1, and to count points in the stack sense,

we need to divide each point by the number of its automorphisms. Thus the

right hand side of (1) is -^-, also.

As so often in mathematics, it turns out the best way to prove Theo-

rem 0.1 is to generalize it. For example, we wish to stratify our stack X and

deduce the theorem for X from the theorem on the strata. This requires that

we consider more general coefficients than Qi (for example, the sheaves one

gets by pushing forward Q^ from the strata). We will also want to perform

various base changes, so we consider a relative version of the theorem (for a

morphism X — 2) instead of just a stack X). This all works very well, if one

has a sufficiently flexible l-adic formalism at ones disposal. Constructing

such a formalism occupied the main part of this paper.

There is a construction of a derived category of equivariant sheaves due

to Bernstein and Lunts (see [5]). This is a topological analogue of a special

case of our derived category. (The equivariant case is the case of quotient

l