of schemes should even induce a continuous functor /* of these corresponding
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
etale sheaves top(X#§t) associated with it. Again, this gives rise to a c-topos
). As above, we consider the derived category
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.
).) The miracle is that D^art(Xm6tlAf) = D~tt(XSTn, Af)
(see Proposition 3.63).
So we can use
to define / * and
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 ) Dbc(X6t,A) =
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