Timeline for A nice explanation of what is a smooth (l-adic) sheaf?
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17 events
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| Jul 23, 2010 at 1:41 | comment | added | BCnrd | Dear Matt: Aha, I didn't know that you had in mind complex algebraic varieties when suggesting the topological case. (See, I still haven't become familiar with what is in Borel's book....sorry.) Certainly you're right on with the Picard-Lefschetz & vanishing cycle stuff, and that for proper base change and so forth the complex algebraic case (with complex topology) is a priceless source of intuition. | |
| Jul 22, 2010 at 18:39 | comment | added | Emerton | ... This applies even more to the theory of nearby/vanishing cycles. Incidentally, Picard--Lefschetz theory goes back to the two people appearing in the name. I think that Lefschetz in particular was instrumental in introducing algebraic topological methods into algebraic geometry. It seems to me that the notions that are being captured in the theory of $\ell$-adic sheaves have their origin in this work, and so I don't think that advocating learning the complex theory prior to the $\ell$-adic theory is particularly ahistorical. | |
| Jul 22, 2010 at 18:36 | comment | added | Emerton | 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. | |
| Jul 22, 2010 at 11:17 | comment | added | Donu Arapura | I'm not a historian, but I think that constructible sheaves arose in alg. geom. first. | |
| Jul 22, 2010 at 2:34 | comment | added | BCnrd | Mikhail, from personal experience you don't need the topological theory of constr. sheaves to learn etale cohom. well. Emerton is right that the loc. constant case (and its relation with $\pi_1$-repns, etc.) is vital source of intuition; for constr. case one can dive right into etale theory. If anything, it's "easier" since stratification in topological case is a more delicate concept (since closed sets can be so bad). Early SGA2 has nice basics with excision-type arguments in topology, good enough to get into etale case. Did constr. sheaves really arise first in topology, not alg. geom.? | |
| Jul 21, 2010 at 13:06 | comment | added | Donu Arapura | Mikhail, I know you're interested in motives. So I purposely included examples that should be "motivic" in nature, but I won't attempt to make this precise. | |
| Jul 21, 2010 at 0:57 | vote | accept | Mikhail Bondarko | ||
| Jul 21, 2010 at 0:57 | comment | added | Mikhail Bondarko | Dear Emerton, I am coming to this subject from a 'motivic' and 'formal' point of view. Yet probably I should find time and study the topological side of the story. | |
| Jul 21, 2010 at 0:52 | comment | added | Emerton | 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. | |
| Jul 20, 2010 at 23:57 | answer | added | Donu Arapura | timeline score: 28 | |
| Jul 20, 2010 at 20:25 | history | edited | Mikhail Bondarko |
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| Jul 20, 2010 at 13:00 | comment | added | BCnrd | 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. | |
| Jul 20, 2010 at 9:53 | comment | added | Mikhail Bondarko | Both references and short explanations would be very welcome! | |
| Jul 20, 2010 at 6:46 | comment | added | Victor Protsak | Is this a reference request? | |
| Jul 19, 2010 at 14:05 | comment | added | Daniel Larsson | I agree with Brian. Let me however add the following reference which might be useful: springer.com/mathematics/algebra/book/…. | |
| Jul 19, 2010 at 13:06 | comment | added | BCnrd | 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. | |
| Jul 19, 2010 at 12:31 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |