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'). ^
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
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