Timeline for Obstruction to lifting coherent sheaves on discrete valuation ring
Current License: CC BY-SA 3.0
10 events
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| Sep 2, 2017 at 4:08 | comment | added | nfdc23 | Minor comment: you should assume $R$ is complete or at least henselian to ensure that its integral closure in each finite extension of $K$ is again local (not just semi-local) and hence $\overline{R}$ is local and thus has a single "residue field" over the one of $R$. | |
| Sep 1, 2017 at 18:15 | comment | added | Piotr Achinger | In general you can imagine that there is a relative moduli scheme (or stack) $\mathcal{M}/R$ parametrizing coherent sheaves on $\mathcal{X}$ flat over the base. Then the question becomes: suppose $\mathcal{M}(\bar R)\to \mathcal{M}(k)$ is surjective (which is close to saying that $\mathcal{M}$ itself is flat), then is $\mathcal{M}(R)\to \mathcal{M}(k)$ surjective as well? The answer is yes if $R$ is complete (or henselian) and $\mathcal{M}$ is smooth over $R$, and you might be able to detect this using the deformation theory of $E$. For example, the vanishing of ${\rm Ext}^2(E, E)$ is enough. | |
| Sep 1, 2017 at 15:15 | history | edited | Chen | CC BY-SA 3.0 |
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| Sep 1, 2017 at 15:02 | comment | added | Chen | @PiotrAchinger What if I substitute coherent by locally free in the above question? | |
| Sep 1, 2017 at 14:57 | comment | added | Piotr Achinger | Sorry, I meant $E$ to be the skyscraper sheaf of $0$, and above I show that it extends to a flat $E_{\bar R}$ but it doesn't extend to an $E_R$. | |
| Sep 1, 2017 at 14:54 | comment | added | Piotr Achinger | Here is a possible counterexample: Let $t$ be a uniformizer of $R$ and let $\mathcal{X}=\{xy = t\} \subseteq \mathbf{A}^2_R$. Let $0$ be the node in the special fiber. One checks easily that there is no section of $\pi$ through $0$. On the other hand, the point $(\sqrt{t}, \sqrt{t})\in \mathcal{X}(\bar R)$ is such a section $s:\bar R\to \mathcal{X}_{\bar R}$ over $\bar R$. Now take $E$ to be the structure sheaf of this section i.e. $E = s_* \mathcal{O}_{{\rm Spec}\, R}$. | |
| Sep 1, 2017 at 13:05 | comment | added | Chen | @LaurentMoret-Bailly Ofcourse. Sorry, I forgot to specify the flatness. I have edited the question. | |
| Sep 1, 2017 at 13:04 | history | edited | Chen | CC BY-SA 3.0 |
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| Sep 1, 2017 at 10:54 | history | edited | Chen | CC BY-SA 3.0 |
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| Sep 1, 2017 at 10:41 | history | asked | Chen | CC BY-SA 3.0 |