Timeline for A reference for Serre duality
Current License: CC BY-SA 4.0
12 events
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| Dec 17, 2021 at 12:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 17, 2021 at 11:16 | answer | added | Leo Alonso | timeline score: 1 | |
| Nov 16, 2021 at 8:05 | comment | added | Sándor Kovács | @Hephaistos, all of this really comes down to that this is a special case of Grothendieck Duality and if you view it that way, then I think all of your questions are answered. In this lemma in particular, the K is allowed to be a complex. As for the restriction question, it is true in general that a dualizing complex restricts to a dualizing complex. I am sure there is a reference for that on the stacks project. | |
| Nov 16, 2021 at 0:03 | comment | added | Hephaistos | Another question that is not treated in Hartshorne's book (except in exercise 7.2) is the local nature of the dualizing sheaf, i.e. if $X$ has one, then for every open subset $U\subset X$, $\omega_{X|U}$ is a dualizing sheaf for $U$. This would be interesting in the case of a non smooth (or even non reduced non quasi-projective) $X$. In this case there is little hope to describe $\omega_X$ in terms of $\Omega_X$. But it would be nice if it was possible to use an open cover $(U_i)$ of $U$, compute the dualizing sheaves of the $U_i$, and compute $\omega_X$ by gluing the $\omega_{U_i}$. | |
| Nov 15, 2021 at 23:54 | comment | added | Hephaistos | @Sándor Kovács I assume (but it is not clear for me) that Lemma 48.27.5 implies that the isomorphisms are fully functorial, i.e. if we have an exact sequence $\Sigma$ of (quasi-)coherent sheaves then the isomorphisms provide an isomorphism between the long exact sequence of $\Sigma$ in cohomology and the dual of that with the Ext's (in other words the isomorphisms of Lemma 48.27.5 commute with $\delta$-operators). | |
| Nov 15, 2021 at 23:47 | comment | added | Hephaistos | @R. van Dobben de Bruyn and @ Sándor Kovács. Thank you for the answers. The missing dual sign has been corrected. | |
| Nov 15, 2021 at 23:46 | history | edited | Hephaistos | CC BY-SA 4.0 |
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| Nov 15, 2021 at 18:46 | comment | added | Sándor Kovács | Also, BTW, @Hephaistos, I think there is a dual sign missing from your displayed line, no? | |
| Nov 15, 2021 at 18:46 | comment | added | Sándor Kovács | I think stacks.math.columbia.edu/tag/0FVZ is a more direct reference for what the OP is asking... | |
| Nov 14, 2021 at 16:16 | comment | added | R. van Dobben de Bruyn | See Tags 0AU9 (existence of $\omega_X^\bullet$ on varieties) and 0AWT (dualising complex on CM lives in one degree). | |
| Nov 14, 2021 at 15:53 | history | edited | Hephaistos | CC BY-SA 4.0 |
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| Nov 14, 2021 at 15:48 | history | asked | Hephaistos | CC BY-SA 4.0 |