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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.

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Lisse (and more generally, constructible) $\ell$-adic sheaves are nicely explained (their properties, link with $\pi_1$-representations, etc.) in the standard references on etale cohomology (Milne's book, Freitag-Kiehl book, SGA4,...), so can you clarify what it is that you wish to understand which is not adequately addressed in such places? Lisse sheaves are important for plenty of things more basic than perverse sheaves as well, so the question seems a bit too brief as presently written. – BCnrd Jul 19 '10 at 13:06
I agree with Brian. Let me however add the following reference which might be useful:…. – Daniel Larsson Jul 19 '10 at 14:05
Mikhail, though lisse sheaves are nicest, it is constructible ones that admit the widest range of operations (e.g., excision, "noetherian" characterization at "finite level", preservation under higher direct images with proper support -- as well as under higher direct images in many important cases, as explained in Deligne's SGA 4 1/2 "Th. finitude..."), so to prove things about lisse sheaves often one needs a version for constructible ones and then apply "specialization criterion" for constructible to be lisse. This comes up in proof of the smooth and proper base change thm, for example. – BCnrd Jul 20 '10 at 13:00
If you are not comfortable with the topological theory of locally constant and constructible sheaves (on varieties over $\mathbb C$, say, as explained in e.g. Borel et al.'s book on Intersection Homology) then I would suggest that you learn this theory first before learning the $\ell$-adic theory. Once you know the former, the $\ell$-adic theory will seem much more motivated (see e.g. Donu Arapura's answer below), and it becomes safe to treat it as a black box to a large extent. On the other hand, if you don't know the former, then the latter will not make much sense. – Emerton Jul 21 '10 at 0:52
Dear Brian, One doesn't have to allow non-Zariski closed stratifications to be in the topological setting (and I'm not advocating that one should). Rather, I am advocating that it makes sense to understand the case of arguments with sheaves (constructible along Zariski closed subsets) in the usual topology on complex varieties before passing to the etale setting. Results such as proper base-change, and local acyclicity of smooth morphisms, are simpler to understand in this setting, and (personally) I find the resulting intuition very important for understanding the etale setting. – Emerton Jul 22 '10 at 18:36

1 Answer 1

up vote 17 down vote accepted

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.

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Thank you! Yet (as far as I remember) people also talk about lisse sheaves on big (etale) sites. Does there exist a similar explanation for this? – Mikhail Bondarko Jul 21 '10 at 0:59
Is $f:Y\to X$ also assumed to be surjective? – Victor Protsak Jul 21 '10 at 2:00
Donu, not obvious that analytification of lisse $\mathbf{Z}_{\ell}$-sheaf, as inv. system of finite loc. constant sheaves on $X(\mathbf{C})$, comes from loc. constant sheaf of finite-rank $\mathbf{Z}_{\ell}$-mods. Need open cover "trivializing" all levels of analytified inv. system. Smooth case OK; o/w need work on topology of $X(\mathbf{C})$. Crucial in pf of relative $\ell$-adic Artin comparison isom (with constr. coeffs), for which there's no published pf! (SGA4 has finite coeffs, SGA5 nada, BBD misses it.) Deligne has nifty triangulation-free pf via alterations (private communication). – BCnrd Jul 22 '10 at 2:27
Brian, thanks for clarifying. It would be nice if someone (hint, hint) wrote something more complete about this. – Donu Arapura Jul 22 '10 at 11:15
Donu, I have written up Deligne's nice argument some time ago. Some day it will appear in something else which I need to get done. – BCnrd Jul 23 '10 at 1:42

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