Let $X$ be an affine scheme with structure sheaf $\mathcal{O}_{X}$. Let $M$ be a finitely generated $\Gamma(X,\mathcal{O}_{X})$-module, say $s_{1},…,s_{n}$ are the generators. We consider the ''tilde-construction" applied to $M$ to obtain an $\mathcal{O}_{X}$-module $\widetilde{M}$. Notice that $\widetilde{M}(X) = M$ and thus is also finitely generated. As a consequence I came up with the following question.
Question: Is the sheaf $\widetilde{M}$ also in some sense finitely generated? For instance, do we have for every open $U\subset X$ that $s_{1}\rvert_{U},…,s_{n}\rvert_{U}$ generate $\widetilde{M}(U)$ as $\Gamma(U,\mathcal{O}_{X})$-module?
Best Answer
The notion you are looking for is for a sheaf $\mathcal{F}$ of $O_X$ modules to be "finitely generated by global sections," meaning there is a collection $\{s_i\}\in\Gamma(X,\mathcal{F})$ so that the images of the $s_i$ generate the stalk $\mathcal{F}_x$ as an $O_{X,x}$-module for every $x\in X$. Here's an exercise for you (it's not difficult, I promise): Let $\mathcal{F}=\widetilde{M}$ on Spec $A$. Show that any set of generators for $M$ as an $A$-module generate $\mathcal{F}$ as an $O_X$ module. Futhermore, if $M$ is finitely generated, so if $\mathcal{F}$.