Timeline for existence of a coherent sheaf
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 19 at 5:05 | vote | accept | KAK | ||
| Oct 19 at 5:04 | vote | accept | KAK | ||
| Oct 19 at 5:05 | |||||
| Oct 18 at 3:02 | history | edited | KAK | CC BY-SA 4.0 |
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| Oct 18 at 3:02 | answer | added | KAK | timeline score: 2 | |
| Oct 17 at 11:00 | comment | added | KAK | @Z.M Ok got it .Thank you | |
| Oct 17 at 10:50 | comment | added | Z. M | I mean, any finitely projective projective module equipped with a surjection. This is how one produces projective resolutions, but since the module in question is coherent, you can pick finitely generated projective resolutions. | |
| Oct 17 at 10:46 | comment | added | KAK | @Z.M Can you please tell how to get the surjection $p$? | |
| Oct 17 at 10:18 | comment | added | Z. M | When the ring $A[t^\pm]$ is coherent, you can pick any finite projective $A[t^\pm]$-module (which is coherent, by coherence of $A[t^\pm]$) along with a surjection $p$ to $\mathcal F(X)$, and the coherence of $A[t^\pm]$ guarantees that the $A[t^\pm]$-module $\ker p$, a priori of finite type, is also coherent. | |
| Oct 17 at 10:02 | comment | added | KAK | @Z.M If A is Noetherian ring Then which resolution i need to use? | |
| Oct 17 at 9:52 | comment | added | Z. M | As you have already observed, the scheme $X=\operatorname{Spec}(A[t^\pm])$ is affine, so you can just formulate it as a problem of modules. However, I am not sure what assumptions you have on $A$. If $A$ is stably coherent (e.g. a Noetherian ring, or a valuation ring), then this simply follows from existence of flat resolutions. | |
| Oct 17 at 5:40 | review | Close votes | |||
| Oct 22 at 3:09 | |||||
| Oct 17 at 5:31 | comment | added | KAK | @Sasha Sorry for the bad mistake. I have edited my question | |
| Oct 17 at 5:30 | history | edited | KAK | CC BY-SA 4.0 |
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| Oct 17 at 5:22 | comment | added | Sasha | In fact $\mathrm{Tor}_1^{\mathcal{O}_X}(G,\mathcal{O}_X) = 0$ for any quasicoherent sheaf $G$, because $\mathcal{O}_X$ is $\mathcal{O}_X$-flat. | |
| Oct 17 at 5:03 | history | asked | KAK | CC BY-SA 4.0 |