2

INTRODUCTION

stacks.) The similarity can be seen from their Appendix B, where a descrip-

tion of their category is given, which is formally close to our category.

We construct an £-adic derived category for stacks, called D^(X, Q^),

where c stands for 'constructive' (we also have versions where c is replaced

by m for 'mixed' and a for '(absolutely) convergent'). We also construct the

beginnings of a Grothendieck formalism of six operations, but we restrict

ourselves to i?/*, /* and Rf', which are the three operations we need to

prove the trace formula.

There are two main problems that one faces if one wants to construct

such an £-adic formalism for stacks. First of all, there are the 'topological'

problems, consisting in finding the correct site (or topos) with respect to

which one defines higher direct images Rf* and pullbacks /* and /*. The

etale topos is certainly too coarse. The second problem is an £-adic prob-

lem dealing with defining a well-behaved £-adic formalism for unbounded

complexes.

Let us describe these problems in more detail.

Topological Problems. We need a cohomological formalism for con-

structive torsion sheaves. Let A' be a ring whose characteristic is invertible

on all algebraic stacks in question. Let X be an algebraic stack. Then we

have the category Mod(Xet, ^4/) of etale sheaves of ^-modules on X. We

wish thf.s category to be the category of coefficient sheaves. Now for alge-

braic stacks the etale topology is not fine enough to compute the correct

cohomology groups of such a sheaf F G obM.od(X&tlA'). ^ f°

r

example,

G is a connected algebraic group and X = BG is the classifying stack of

G, then BG^ — S^t, where S is the base we are working over (see Corol-

lary 4.30). The stack BGmi for example, should have the cohomology of

the infinite dimensional projective space, but from the etale point of view

BGm looks like a point. This leads us to consider the smooth topology Xsm

on the stack X. We use the smooth topology to compute cohomology of

etale coefficient sheaves. Thus we are dealing with a pair of topoi (or sites)

(Xsm, X^t). Since this situation arises in other contexts as well, we formalize

it axiomatically. We call such pairs of topoi c-topoi (see Section 4).

Thus we consider the derived category

D(XsmiAf)

of the category of

smooth sheaves of A'-modules and pass to the subcategory

D^t(XsmiAf)^

defined by requiring the cohomology sheaves to be etale. This gives rise

to the definition of Rf* : Dft(Xsm,A') — D^(2)

s m

, A'), for a morphism

of algebraic stacks f : X — 2). If X = X is a scheme, then we have

D+(Xam,A') = D+{Xst,A') (Proposition 3.42).

One of the problems with the smooth topology is the correct choice of a

site defining it. Since all the problems arise already for schemes, let X be a

scheme. One possible definition of a smooth site would be to take all smooth

X-schemes with smooth morphisms between them. The problem with this

definition is that products in this category are not what they should be, if

they even exist at all. (For a smooth morphism to have a smooth diagonal it

has to be etale.) This means that it is not clear why a morphism / : X — Y