A nice explanation of what is a smooth (l-adic) sheaf?

Question

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.

Answer 1

I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale site $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). For lisse sheaves, each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy. Remark added July 22: Unfortunately, this part of the story appears to be inadequately addressed in the literature. See BCnrd's comment below.