Timeline for Which sheaves on a projective bundle are flat over the base scheme?
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Oct 5, 2015 at 16:22 | comment | added | Bernie | Thank you both. The hint that such $G$ are of the form $f^{*}H$ was what I needed. This solves my problem completely. If someone will write a short answer, then I can accept it. | |
| Oct 5, 2015 at 15:42 | comment | added | t3suji | Also, if you are just interested in sheaves such that $f^*f_*G\to G$ is an isomorphism, you can conclude that there are no higher direct images without needing flatness (in this situation). Namely, such sheaves are exactly sheaves of the form $f^*H$ for some $H$ (not necessarily locally free), and use the projection formula. | |
| Oct 5, 2015 at 15:03 | comment | added | Jason Starr | There are many coherent sheaves on $\mathbb{P}(E)$ that are flat over $X$ yet not locally free. For instance, if $X$ is $\text{Spec}(\mathbb{C})$ itself, then every coherent sheaf on $\mathbb{P}(E)$ is flat over $X$. In another direction, for every section of $f$, $s:X\to \mathbb{P}(E)$, the coherent sheaf $s_*\mathcal{O}_X$ is flat over $X$. However, if a coherent sheaf $G$ is flat over $X$ and $f^*f_*G\to G$ is an isomorphism, then $G$ is of the form $f^*H$ for a locally free sheaf $H$ on $X$. | |
| Oct 5, 2015 at 10:03 | history | asked | Bernie | CC BY-SA 3.0 |