Timeline for Leray-Hirsch principle for étale cohomology
Current License: CC BY-SA 2.5
18 events
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| Sep 20, 2010 at 20:09 | comment | added | Torsten Ekedahl | Dear Algori, note that I have answered your last question in a revision (though not exactly in the way you suggested but rather by proving directly the stacky statement). | |
| Sep 2, 2010 at 6:55 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added details on claimed statement.
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| Aug 31, 2010 at 14:52 | comment | added | algori | Dear Torsten -- I have a question regarding the "stacky" procedure: how does one prove that the map from the stack quotient to the quotient induces a cohomology isomorphism? Basically, this boils down to proving the Leray-Hirsch theorem, but this time one can assume that all geometric fibers are acyclic say with $\mathbf{Q}_l$ coefficients, but I'm not sure what exactly this implies since the map is not proper. | |
| Jun 28, 2010 at 7:48 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Typo
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| Jun 25, 2010 at 23:11 | vote | accept | algori | ||
| Jun 25, 2010 at 18:44 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Proved also the smooth case
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| Jun 24, 2010 at 6:04 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
deleted 2 characters in body
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| Jun 24, 2010 at 6:00 | comment | added | Torsten Ekedahl | (cont'd) If one uses the condition that I use in my present version, i.e., that the fibre of the direct images are the cohomology of the fibres, then I don't know of any topological version of the LH result that is not covered. | |
| Jun 24, 2010 at 5:59 | comment | added | Torsten Ekedahl |
@Dustin: I was a little bit hasty in the smooth case and have made a retraction. Note however that your case of $\mathbb A^2\setminus\{0\}\to \mathbb A^1$ does not fulfil the LH-condition. I still have some hope that given LH condition things would be OK. For the case of a torus quotient I agree that in general it can be tricky but given the LH condition I think it should be easier. @algori: I am not sure what you mean by a "statement similar to the topological case".
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| Jun 24, 2010 at 5:52 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Reorganised and retracted one statement
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| Jun 24, 2010 at 2:49 | comment | added | algori | Torsten, thanks again! Re your second proposal: how does one compute the cohomology of a torus quotient? This can be quite tricky even in the complex case. All in all it seems that the stacky way would be the less painful one. Is there a reference where this (or any of the other possible approaches you describe) has already been implemented? But still, somehow I believe that this has nothing to do with group actions and that there is a statement similar to the one in the topological case. | |
| Jun 23, 2010 at 21:21 | comment | added | Dustin Clausen | Dear Torsten, I was wondering if you might help me understand your comment about the fibers of p_*A when p is smooth, since I'm having trouble picturing what you're saying. To my eye, your claim implies that the pushforwards of the constant sheaf along A^2 --> A^1 and A^2 \ 0 --> A^1 will agree, which doesn't square. What am I missing? | |
| Jun 23, 2010 at 20:16 | comment | added | Torsten Ekedahl | OK, I've added a section which should cover that case. | |
| Jun 23, 2010 at 20:14 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added other conditions giving the conclusion
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| Jun 23, 2010 at 16:07 | comment | added | algori | Torsten -- an typical example would be a smooth affine variety on which a reductive group acts with finite stabilizers. Then the geometric quotient exists and we can then consider the map into the quotient. As far as I understand this is not a fibration (or is it?) In any case the map is not proper. However in some cases one can construct the "Leray-Hirsch" cohomology classes. | |
| Jun 23, 2010 at 14:56 | comment | added | Torsten Ekedahl | 1. This is what my argument was supposed to show, I can expand if you point out what part is unclear to you. 2. As I said if $p$ is proper you are always OK otherwise I might be a little bit tricky however there are quite a few situations when it is still OK. On the other hand most of the classical cases where one applies LH have algebraic analogues which are fibrations in the étale topology so I would need more specificity to say more. | |
| Jun 23, 2010 at 13:47 | comment | added | algori | Thanks, Thorsten! A couple of remarks: 1. How exactly does one show in the etale case that the derived direct image decomposes as a direct sum of shifted constant sheaves? 2. I really would like to avoid assuming $p$ a fibration in the \'etale topology. | |
| Jun 23, 2010 at 5:00 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |