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
Previous Page Next Page