# The category of l-adic sheaves

I'm currently trying to understand the construction of the category of l-adic constructible sheaves as in SGA5, and it seems that quite a lot of machinery (the MLAR condition, localization of the category of projective systems, etc.) has to be gone through before one can even construct this category and show that it's abelian, for instance. On the other hand it is not even true that the derived category of l-adic sheaves is defined in the obvious manner, since it is defined as a 2-limit of the derived categories of $\mathbb{Z}/l^n$ constructible sheaves.

I understand that the categorical machinery in existence today is a lot more powerful than it was in the 1970's, which makes me curious: is there a cleaner and more transparent way of doing this, and a more modern presentation than SGA or Frietag-Kiehl?

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Other than Ekedahl, "On the adic formalism", I recommend Laszlo-Olsson, "The six operations..., I, II", which made use of Gabber's recent result on finiteness in étale cohom. Their results are more general in that: 1. stacks, 2. over a base more general than a field with finite l-cd and 3. for unbounded derived categories. Note that SGA5 didn't do derived category: just l-adic sheaves. Same issue with SGA4.5 Rapport and Th. finitude; note that under the title "Th. de finitude en cohomologie l-adique" the results are not stated for adic case, but of course this happened after SGA5. –  shenghao Jun 28 '11 at 20:36
Thanks for the references! –  Akhil Mathew Jun 29 '11 at 0:52
Not sure how helpful this is, but I'm pretty sure the constructions in SGA only give the "correct" answer when the ground field is either finite or algebraically closed. So one might hope not only for a "cleaner and more transparent way" but also greater generality. –  Justin Campbell Nov 12 '11 at 18:46

Section 1.4 in these notes of Brian Conrad is as nice as one could hope for, given the dryness of the adic formalism. I don't think the material differs substantially from Frietag-Kiehl, except in that the presentation is much cleaner. For the derived category stuff, the notes refer to Behrend's paper "Derived $\ell$-adic Categories for Algebraic Stacks" which I haven't looked at really, but a brief skim suggests it contains everything you might desire (constructions in extremely general situations), but nonetheless includes examples (!).

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If you ask Behrend for giving you an example, it will end up in sort of new prime numbers, which will be called Behrend's prime numbers! –  Ehsan M. Kermani Nov 12 '11 at 10:06
Thanks. I had seen Conrad's notes earlier, but not Behrend's; this is very nice. –  Akhil Mathew Nov 12 '11 at 13:43
I'm just going to point out, incidentally, that the (derived?) $\infty$-category of $\mathbb{Z}_l$-sheaves is apparently the homotopy limit (in the Joyal model structure) of the category of $\mathbb{Z}/l^n$-sheaves. This is one thing that I'd like to understand eventually, but I haven't seen it written up anywhere. –  Akhil Mathew Nov 12 '11 at 13:44
@Akhil: Perhaps one would prove that (or, say, the $\mathbb{Q}_{\ell}$-analogue) by showing both sides are the derived categories of perverse sheaves, the classical definition having been done by Beilinson and the $\infty$-categorical version amenable to a similar treatment. But it would definitely be nice to have it written up. The limit Deligne takes appears (to me) to work only out of "good luck," (since it's not a real operation in the world of triangulated categories) whereas the homotopy limit is much more natural. –  Moosbrugger Nov 12 '11 at 16:26
@Moosbrugger: I'm intrigued by this refinement of Beilinson's theorem, but have never heard it before. (It seems, though, to construct the abelian category perverse sheaves, one must first construct the derived category of $l$-adic sheaves the normal way -- so it would be some kind of bootstrapping.) –  Akhil Mathew Nov 13 '11 at 1:52

Zheng and Liu are using $\infty$-categories to study constructible sheaves on stacks, and they have a $\ell$-adic version too. (Though most of the details for the $\ell$-Adic version should appear in a second paper that is still in preparation, and I would not call their first paper easy to read. But it is certainly a modern presentation...) Reference : http://math.columbia.edu/~zheng/bc1.pdf

By the way, they use Gabber's finiteness results, and there is now a nice reference for these too ! (This is really cool.) http://www.math.polytechnique.fr/~orgogozo/travaux_de_Gabber/

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Very interesting. Thanks for these links. –  Akhil Mathew Apr 10 '12 at 6:19