Timeline for "Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology
Current License: CC BY-SA 3.0
17 events
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| Oct 20, 2016 at 21:59 | vote | accept | CommunityBot | ||
| Oct 7, 2016 at 7:41 | answer | added | ACL | timeline score: 9 | |
| Oct 7, 2016 at 7:17 | comment | added | user40276 | @AlexYoucis I'm just trying to make the analogy precise. Maybe I'm totally wrong. About an abstract notion of 6 functor formalism, maybe something like definition A.1.10 of arxiv.org/pdf/1305.5361.pdf would be enough (with some modifications, like dropping Poincare duality or forgetting the Tate twists). About these being in some sense the 'six operation hull', I think it's not exactly true. For instance, perverse sheaves have analogous properties but they're the heart of a non-canonical t-structure in constructible sheaves. Thinking better now, this analogy seems to be a little forced | |
| Oct 7, 2016 at 6:59 | comment | added | HeinrichD | Since this question has gained 13 upvotes in 13 hours, I assume that many people are here who understand and appreciate it, but actually I don't understand what is being asked here (although I am familiar with the concepts of cohomology and coherent sheaves). Could someone explain this to me, in his own words? If I take the question literally, I don't know what to say except for that both notions of cohomology are actually the same, just applied to different sites. | |
| Oct 7, 2016 at 6:51 | comment | added | Alex Youcis | meaning the smallest subcategories closed under the six operations and containing an 'important subcategory' (vector bundles for Coh, and lisse Z_\ell sheaves for constructible)? | |
| Oct 7, 2016 at 6:50 | comment | added | Alex Youcis | @user40276 I see--I suppose that makes sense, but I have no idea how you deduced that from the OPs question or their above comment. Anyways, is there actually a 'six functors formalism' in the abstract? Namely, as far as I know there's no abstract definition of a 'six functors formalism' to make precise the statement 'Qcoh is the largest subcategory of Ab having a 'six functors formalism' '. Is there one I am unaware of, or were you just giving an intuition about what sort of thing the OP wants? Isn't it true that in both Qcoh and constructible they are the 'six operations hull' | |
| Oct 7, 2016 at 6:42 | comment | added | user40276 | @AlexYoucis I think the OP means that there's a deep analogy between coherent (maybe quasi-coherent of finite presentation) sheaves in Zariski topology and constructible (or maybe lisse, or torsion) l-adic sheaves in the étale topology. And he\she wants to make it precise. For instance, there's a Verdier duality in both cases and a six functor formalism. Maybe a precise answer would be something like a six functor formalism only exists for some kind of sheaves in a given topology such that for Zariski it would be the coherent ones and for étale it would be the constructible ones. | |
| Oct 7, 2016 at 5:59 | comment | added | Alex Youcis | in the etale setting. I apologize if I have misinterpreted your question. Dorebell seems to think you're asking for the analogous objects in etale cohomology for coherents in coherent(=Zariski) cohomology. If that's true then, yes, I agree that for most applications of etale cohomology one is usually interested in the cohomology of a lisse $\mathbb{Z}_\ell$-sheaf (or perhaps a constructible $\mathbb{Z}_\ell$-sheaf). | |
| Oct 7, 2016 at 5:57 | comment | added | Alex Youcis | Are you asking about coherent sheaves for the etale topology? One of the first thing one shows (using faithfully flat descent) is that, essentially, $\mathrm{QCoh}(X_{\acute{e}\text{t}})\cong \mathrm{QCoh}(X_{\mathrm{Zar}}$ in the obvious way (namely if $\mathcal{F}$ is quasi-coherent on $X_{\acute{e}\text{t}}$ then its pullback to the Zariski site is a quasi-coherent) and that this preserves global sections. In particular, the etale cohomology is the same as the Zariski cohomology of coherent sheaves. In other words NOTHING new is gained by thinking about quasi-coherent sheaves | |
| Oct 7, 2016 at 2:45 | comment | added | dorebell | In my novice understanding: I think the usefulness of studying cohomology of coherent sheaves on a scheme is primarily for analogy (although there are bound to be some direct applications of the theory). Coherent sheaves are the class of sheaves where cohomology works 'nicely' in the setting of sheaves of modules on a (separated and Noetherian, say) scheme. There are analogous classes of sheaves ($\ell$-adic constructible sheaves, etc.) for the étale cohomology which are the 'nice' ones in that setting. Many similar formal properties hold, and the derived functor formalism unites the two. | |
| Oct 7, 2016 at 1:29 | history | edited | user97565 | CC BY-SA 3.0 |
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| Oct 7, 2016 at 1:18 | history | edited | user97565 | CC BY-SA 3.0 |
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| Oct 7, 2016 at 0:50 | comment | added | user97565 | @user40276 I'm not quite sure myself, but multiple people have said having intuition wrought out of a good grasp of how stuff works in the classical setting a la cohomology of coherent sheaves carries over well to étale cohomology, and I'm hoping someone can talk about that. Maybe I'm way wrong. | |
| Oct 7, 2016 at 0:49 | history | edited | user97565 | CC BY-SA 3.0 |
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| Oct 7, 2016 at 0:44 | comment | added | user40276 | Excuse me, but could you give an example of the usage of coherent sheaves in étale cohomology. I'm a total ignorant, but, as long as I know, only torsion étale sheaves or more generally mixed l-adic sheaves are used at all... | |
| Oct 6, 2016 at 22:22 | history | edited | user97565 | CC BY-SA 3.0 |
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| Oct 6, 2016 at 17:56 | history | asked | user97565 | CC BY-SA 3.0 |