Timeline for Hilbert's syzygy theorem in the analytic setting
Current License: CC BY-SA 3.0
8 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 12, 2014 at 17:08 | comment | added | user76758 | @Eric: Since I assume you have read the proof of GAGA, and hence must know the analytic properties of $O(1)$ and the basic theory of coherent-analytic sheaves (without which that proof doesn't work), you just use exactly the same proof as in the algebraic case. There is probably no reference because it is literally the same argument. Just talk with anyone in algebraic geometry or number theory on the faculty at your university if you have further questions about it. | |
| Jan 12, 2014 at 4:44 | comment | added | Eric O. Korman | @user76758 thanks. maybe you should make your comment an answer. also, do you have good references for these statements? thanks. | |
| Jan 9, 2014 at 20:22 | comment | added | user76758 | The analytic input into the proof of GAGA provides everything which is required, so GAGA isn't needed and the algebraic proof (appropriately formulated) literally works in the analytic setting. Namely, by the analytic ampleness properties of $O(1)$, any coherent analytic sheaf on a closed complex-analytic subspace $X$ of $\mathbf{CP}^n$ has a resolution by vector bundles. So if $X$ is smooth then by Serre's theorem on finiteness of global dimension of regular local rings (applied to stalks of $O_X$) this resolution has vector bundle kernel at the $(\dim X)$-th step. | |
| Jan 9, 2014 at 20:01 | comment | added | agt | Dear Eric, I added a true link towards your question on Math.SE, I hope you do not mind. | |
| Jan 9, 2014 at 19:59 | history | edited | agt | CC BY-SA 3.0 |
I added a true link to the question on Math.SE
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| Jan 9, 2014 at 19:58 | comment | added | Steven Landsburg | You want to add the word "smooth" in the first sentence. | |
| Jan 9, 2014 at 19:53 | history | asked | Eric O. Korman | CC BY-SA 3.0 |