Lemma 35.7.4. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_ X$-module.
Proof. By Lemma 35.7.1 we see that $\mathcal{F}$ is of finite type. Hence Nakayama's lemma (Algebra, Lemma 10.20.1) implies that $\mathcal{F}$ is generated by $r$ sections in the neighbourhood of a point $x \in X$ if and only if $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) \leq r$. Choose an $i$ and a point $x_ i \in X_ i$ mapping to $x$. Then $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) = \dim _{\kappa (x_ i)} (f_ i^*\mathcal{F})_{x_ i} \otimes \kappa (x_ i)$ which is $\leq r$ as $f_ i^*\mathcal{F}$ is locally generated by $r$ sections. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: