Timeline for Dualizing sheaf
Current License: CC BY-SA 3.0
16 events
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| Oct 13, 2016 at 23:16 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Sep 15, 2012 at 20:11 | comment | added | Sándor Kovács | @Yuhao: I just noticed, that this is actually noted in the original answer... | |
| Sep 15, 2012 at 20:10 | comment | added | Sándor Kovács | $X$ is G1 if it is Gorenstein in codimension 1. This is similar to R1 for regular. In other words, there exists a closed set $Z\subset X$ such that $\mathrm{codim}_X Z\geq 2$ and $X\setminus Z$ is Gorenstein. | |
| Sep 15, 2012 at 3:50 | comment | added | Yuhao Huang | What does G1 mean? | |
| Dec 7, 2010 at 22:08 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 4, 2010 at 3:46 | comment | added | Karl Schwede | Sandor, that's a good point. The argument that $\mathcal{H}om_X(\omega_X, \omega_X) \cong \mathcal{O}_X$ is substantially easier than what I wrote when $X$ is G1 (which implies that $\omega_X$ is a line bundle on a big open set). | |
| Dec 3, 2010 at 20:33 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 3, 2010 at 20:22 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 3, 2010 at 19:52 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 3, 2010 at 16:43 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 3, 2010 at 16:41 | comment | added | Sándor Kovács | Dear Brian: you are right (as always). I wrote that in haste and as Hailong and Karl correctly guessed I was thinking of the canonical sheaf in case $X$ is not CM. Of course, it would be wrong to call that the dualizing sheaf. Although, I guess one might consider that in that case nothing deserves that name, so the abuse was not extremely bad (just simply bad). :) | |
| Dec 3, 2010 at 16:37 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Dec 3, 2010 at 15:12 | comment | added | Karl Schwede | Sandor was certainly working on the geometric setting (ie, variety) where dualizing complexes exist, and then by $\omega$ he meant the first non-zero sheaf as BCnrd suggested. This G1 + S2 hypothesis is pretty common when one wants to work on non-normal varieties but still have a ``canonical divisor''. | |
| Dec 3, 2010 at 14:00 | comment | added | Hailong Dao | Perhaps Sandor meant the canonical sheaf, which can be defined as long as $X$ is embeddable in a Gorenstein scheme? | |
| Dec 3, 2010 at 7:09 | comment | added | BCnrd | Dear Sandor: if $X$ isn't CM, so the dualizing complex isn't concentrated in a single degree (on each connected component), then what is meant by $\omega$? Its top nonzero homology sheaf? | |
| Dec 3, 2010 at 6:51 | history | answered | Sándor Kovács | CC BY-SA 2.5 |