INTRODUCTION

3

of schemes should even induce a continuous functor /* of these corresponding

sites.

On the other hand, there is the big smooth site. None of the 'topological'

difficulties arise in this context, but unfortunately, the direct image functor

is not compatible with the etale direct image. It is easy to construct a

morphism / : X — Y of schemes and an etale sheaf F o n I , such that f*F

(with respect to the big smooth site) is not etale. So using the big smooth

site would not generalize the etale theory for schemes.

So one is forced to use the intermediate notion, where we require objects

of the smooth site to smooth over X , but let morphisms be arbitrary. This

has the unpleasant side effect that this construction does not commute with

localization. For every algebraic stack we get a whole collection of relative

smooth sites (see Section 3). A more serious drawback is that, even though

/* is continuous, it is not exact. Thus / : X — * Y does not induce a mor-

phism of the induced smooth topoi, only what we call a pseudo-morphism

of topoi (see Remark 3.32). Typical counter-examples are closed immer-

sions (see Warning 4.42). So the smooth approach does not give rise to a

definition of pullback functors /*.

This phenomenon necessitates a second approach, the simplicial one.

We choose a presentation X — X of our algebraic stack X which gives

rise to a groupoid X\ =£ Xo, where Xo = X and X\ = X xx X. Setting

Xp = X x% ... X£ X defines a simplicial scheme X., which has a category of

P+i

etale sheaves top(X#§t) associated with it. Again, this gives rise to a c-topos

(top(X.^

t

),X^

t

). As above, we consider the derived category

D(X96t1Af)

with the corresponding subcategory Dcart(X96t,Af), defined by requiring the

cohomology sheaves to be cartesian. (Note that X& is the category of carte-

sian objects of top(X.

6 t

).) The miracle is that D^art(Xm6tlAf) = D~tt(XSTn, Af)

(see Proposition 3.63).

So we can use

D^rt{X%4it)Af)

to define / * and

Dft(Xsm,Af)

to define

Rf*. Another miracle is that the functors thus obtained are adjoints of each

other. This is due to the fact that there is just enough overlap between

the two approaches. The simplicial approach works also for Rf* for rep-

resentable morphisms, whereas the smooth approach works for /* if / is

smooth. These two cases are essentially all that is needed by a devissage

lemma (Proposition 4.16).

l-adic Problems. Now let A b e a discrete valuation ring (like Z^), whose

residue characteristic is invertible on all algebraic stacks we consider. Denote

a generator of the maximal ideal of A by L Then to define an ^-adic derived

category of schemes one defines (see Proposition 2.2.15 of [4]) Dbc(X6t,A) =

proj lim

n

D^

t f

(X^

t

,yl/^

n + 1

). This approach depends crucially on the fact

that the complexes involved are bounded. But in dealing with algebraic

stacks we cannot make this convenient restriction. As noted above, the