Timeline for On the dualizing sheaf of a curve
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2015 at 11:54 | vote | accept | user46578 | ||
| Feb 17, 2014 at 16:50 | vote | accept | user46578 | ||
| Feb 17, 2014 at 17:12 | |||||
| Feb 16, 2014 at 21:01 | comment | added | user46578 | @Arapura: Thank you for the answer. I am most interested in the case when $C$ is not reduced. I have read the above mentioned result in the reduced case although they did not give any reference. So, I wanted to know how general the result actually is. | |
| Feb 16, 2014 at 20:51 | comment | added | Donu Arapura | From another point of view, you can think of this as a generalization of the classical adjunction formula $K_C=K_X©|_C$ when $C$ is smooth. | |
| Feb 16, 2014 at 20:43 | comment | added | user46578 | @user76758: Thank you for the answer. This is what I am looking for. | |
| Feb 16, 2014 at 20:39 | comment | added | user76758 | A more general fact (which can be generalized further) is that if $j:Z \hookrightarrow X$ is a closed immersion between Cohen-Macaulay projective schemes over a field $k$, with respective pure dimensions $d \le n$ then $\omega_{Z/k}=\mathcal{Ext}^{n-d}_X(O_{Z}, \omega_{X/k})$. The key is duality for finite morphisms beyond the finite flat case. The derived category framework (as in Hartshorne's R&D book) illuminates these constructions tremendously (and allows one to study the dualizing sheaf locally, which is ill-suited to the viewpoint of characterization by just a "global" property). | |
| Feb 16, 2014 at 19:51 | answer | added | Piotr Achinger | timeline score: 6 | |
| Feb 16, 2014 at 19:46 | comment | added | abx | Well, this supposes some basic knowledge of duality theory. I suggest Altman and Kleiman Introduction to Grothendieck duality theory. | |
| Feb 16, 2014 at 19:37 | history | edited | user46578 | CC BY-SA 3.0 |
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| Feb 16, 2014 at 19:28 | history | asked | user46578 | CC BY-SA 3.0 |