Timeline for More questions about Verdier duality (and related math)
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12 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 26, 2011 at 5:55 | comment | added | Torsten Ekedahl | Yes, that's it. | |
| Jun 24, 2011 at 12:55 | comment | added | James D. Taylor | I guess that's what you meant: to do the cohomology version of Poincare, and then to use the universal coefficient formula to get the usual (homology on one side, cohomology on the other) version. | |
| Jun 24, 2011 at 12:44 | comment | added | James D. Taylor | Okay, so you pointed out two problems: one, the dualizing sheaf should be $K[n]$ rather than $K[-n]$, and the other that my formula for the homology is wrong. So can this be interpreted as a proof of the cohomology version of Poincare duality: $H^k(X,K)=[K,K[k]]=$by my interpretation of the dualizing sheaf $=[K[k],K\otimes K[n]]=[K[k],K[n]]=[K,K[n-k]]=H^{n-k}(X,K)$? Or is there a fault in that proof? | |
| Jun 24, 2011 at 5:12 | answer | added | Greg Friedman | timeline score: 2 | |
| Jun 24, 2011 at 5:10 | comment | added | Torsten Ekedahl | Sorry, didn't look closely at what you said. You are also wrong about the homology, your formula for $H_k(X,\mathbb E)$ couldn't be OK as homology should be covariant in $\mathbb E$. The Verdier version of Poincaré duality is usually formulated purely in terms of cohomology just as is Serre duality. To get the usual topological version one combines this with the universal coefficient formula. | |
| Jun 24, 2011 at 4:44 | comment | added | James D. Taylor | And even if by magic that were true, my interpretation of the dualizing sheaf would be wrong... (otherwise cohomology would be periodic) | |
| Jun 24, 2011 at 4:36 | comment | added | James D. Taylor | If $D_X$ were $K[n]$ in the Poincare case, it still wouldn't make sense... I guess the only thing that would would be $H^k(X,K) \cong [K[k],K[n]]$, but I don't know how to make that fit with anything... | |
| Jun 24, 2011 at 4:24 | comment | added | Torsten Ekedahl | You should not have different signs in the shifts for Serre and Poincaré duality, in fact the shifts should be exactly the same. The conventions used for shifts (whether a positive shift is to the right or the left) is confusing (at least to me) and my guess is that you have found sources that uses different conventions (which is wrong I think there is only one generally accepted convention only I can never remember which it is). | |
| Jun 24, 2011 at 4:02 | history | edited | James D. Taylor | CC BY-SA 3.0 |
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| Jun 24, 2011 at 2:37 | history | edited | James D. Taylor | CC BY-SA 3.0 |
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| Jun 24, 2011 at 2:21 | history | asked | James D. Taylor | CC BY-SA 3.0 |