The Stacks project

Lemma 28.16.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_ X$-modules $\widetilde M$ is a finite type $\mathcal{O}_ X$-module if and only if $M$ is a finite $R$-module.

Proof. Assume $\widetilde M$ is a finite type $\mathcal{O}_ X$-module. This means there exists an open covering of $X$ such that $\widetilde M$ restricted to the members of this covering is globally generated by finitely many sections. Thus there also exists a standard open covering $X = \bigcup _{i = 1, \ldots , n} D(f_ i)$ such that $\widetilde M|_{D(f_ i)}$ is generated by finitely many sections. Thus $M_{f_ i}$ is finitely generated for each $i$. Hence we conclude by Algebra, Lemma 10.23.2. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 28.16: Characterizing modules of finite type and finite presentation

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01PB. Beware of the difference between the letter 'O' and the digit '0'.