Timeline for Local freeness of $\pi_*F(r)$ from flatness of $F$
Current License: CC BY-SA 4.0
9 events
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| Jul 1, 2020 at 2:46 | comment | added | user267839 | yes of course you are right, the proof in 9.9 solves the problem. sorry, as I wrote my last comment I didn't took a glance into theorem's proof of 9.9, only into it's statement. | |
| Jul 1, 2020 at 0:37 | comment | added | R. van Dobben de Bruyn | I also think the (very short) proof of Thm III.9.9 (i)$\Rightarrow$(ii) applies verbatim in this context, because that direction uses neither that $\mathscr F$ is the structure sheaf nor that $T$ is integral. In fact the first is not even stated as an assumption, and the second is only used for (iii)$\Rightarrow$(ii). | |
| Jul 1, 2020 at 0:09 | comment | added | R. van Dobben de Bruyn | You're right, the reference doesn't cover the precise statement, but as @Mohan points out Thm III.12.11 does (together with Thm III.8.8). | |
| Jun 30, 2020 at 23:03 | comment | added | Mohan | This can be deduced from semicontinuity theorem . For sufficiently large $N$, higher direct images of $F(N)$ vanish. Then the direct image is locally free. | |
| Jun 30, 2020 at 22:25 | comment | added | user267839 | Another more important problem is how can we obtain a closed subscheme $X \subset \mathbb{P}^n_T$ in the thereom from coherent sheaf $F$ on $\mathbb{P}^n_S$? $F$ induces a relative spectrum (a candidate for closed subscheme $X$ in the theorem) $X_F:=\operatorname{Spec}_S(F) \to S$ in well known way but is $X_F \subset \mathbb{P}^n_T$? $F$ is by definition only a $O_{\mathbb{P}^n_S}$-module not a $O_{\mathbb{P}^n_S}$-algebra. that is a priori there is no map $O_{\mathbb{P}^n_S}=O_S[x_0,...x_n] \to F$ corresponding to $X \subset \mathbb{P}^n_T$. Do you know how relate $X$ and $F$? | |
| Jun 30, 2020 at 22:16 | comment | added | user267839 | @R.vanDobbendeBruyn: You mean that one which starts with "Let $T$ be an integral noetherian scheme. Let $X \subset \mathbb{P}^n_T$ be a closed subscheme... And the claim is that $X$ is flat over $T$ iff Hilbert polynomial $P_t$ is independent of $t$. Here is required that $T$ is integral, especially reduced, the book seems to work without it but thats subtile. | |
| Jun 30, 2020 at 21:43 | comment | added | R. van Dobben de Bruyn | See for example Hartshorne, Theorem III.9.9. The idea is to use a Čech resolution and Serre vanishing. | |
| Jun 30, 2020 at 21:42 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jun 30, 2020 at 21:25 | history | asked | user267839 | CC BY-SA 4.0 |