Timeline for Is there an easy proof of the fact that the intermediate image functor respects weights?
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| Feb 8, 2011 at 16:46 | history | edited | Alex | CC BY-SA 2.5 |
Added new thoughts on the question.
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| Feb 6, 2011 at 23:11 | comment | added | Mikhail Bondarko | Thank you very much for your comments! I wasn't offended at all. I just wondered whether you had in mind some piece of the study of motives that I don't know of (or forgot:)). Also, to me my own results seem to be very much easier than those of Deligne.:) | |
| Feb 6, 2011 at 22:55 | comment | added | Alex | And what I meant is more natural for motives is the fact that $f_*$ and $f^!$ increase weights, while $f_!$ a,d $f^*$ decrease weights, when you define weights the way you do in your papers, that is (if I understand well) by declaring that relative Chow motives have weight $0$ and that shifting a motive shifts its weight (I am changing the signs of your weights because that is what I am used to). What of course is harder is the analogue of BBD 5.3.1 because we don't even know how to define a $t$-structure on the triangulated category of motives. But I imagine you know all that better than me. | |
| Feb 6, 2011 at 22:49 | comment | added | Alex | Okay, the problem is that 5.3.1 (if a perverse sheaf is of weights between $a$ and $b$, then so is any subobject) is crucial, and I do not see a more direct way to get at it than the one in BBD. The definition of weights by looking at stalks is not well adapted to perverse sheaves... About motives, I should not have written "easy", because very few things are easy in maths, and I realized that I called part of your work easy although I do not think it is, so I did not mean any offense. I meant something like "the proof seems more natural to me" (more natural than the proof of Weil II). | |
| Feb 6, 2011 at 21:02 | vote | accept | Mikhail Bondarko | ||
| Feb 6, 2011 at 20:47 | comment | added | Mikhail Bondarko | Well, I do not know for sure which sort of a reasoning could help me.:) I would prefer one that relies on 5.1.14 of BBD. Could you say more about what is easy to prove for motives? | |
| Feb 6, 2011 at 20:18 | comment | added | Alex | ...but very easy to prove for motives. And some of these properties are still conjectural for motives. Note also that your statement is corollary 5.4.3 of BBD, which is deduced in BBD from theorem 5.4.1. So I was actually following BBD afer all. :) As for your original question, if what you want is a proof that is very simple and uses only the definition of weights of $\ell$-adic complexes but not, say, Weil II (which is very hard to prove), then I know of no such proof. Sorry. | |
| Feb 6, 2011 at 20:16 | comment | added | Alex | You did not ask for a proof that follows BBD, you asked for a proof that uses the natural properties of $j_{!*}$ and the formal properties of weights. But none of the properties of weights are easy to prove for $\ell$-adic complexes, so there are no formal properties of weights. I think the best you can hope for is a proof that uses the natural properties of weights. The precise proof of those natural properties is going to depend on your particular context (and on your definitions). For example some of those properties are very hard to prove in the $\ell$-adic world... | |
| Feb 6, 2011 at 19:58 | comment | added | Mikhail Bondarko | Your reasoning is certainly very logical; yet it does not seem to follow BBD. In BBD 5.3.2 is not deduced from 5.4.1. Conversely, 5.4.1 relies on 5.3.7, which uses 5.3.5, which uses 5.3.4, and the latter is a corollary from 5.3.2.:) And 5.3.2 is deduced from 5.3.1, which relies on 5.2.1, whereas the proof of the latter is pretty complicated.:) | |
| Feb 6, 2011 at 18:19 | history | answered | Alex | CC BY-SA 2.5 |