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 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
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