The Stacks project

Lemma 32.20.2. Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{ s\} \to S$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \setminus \{ s\} $ there is an equivalence of categories

\[ \left\{ \mathcal{O}_ S\text{-modules }\mathcal{F}\text{ of finite presentation} \right\} \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha ) \right\} \]

where on the right hand side we consider triples consisting of a $\mathcal{O}_ U$-module $\mathcal{G}$ of finite presentation, a $\mathcal{O}_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})}$-module $\mathcal{H}$ of finite presentation, and an isomorphism $\alpha : \mathcal{G}|_ V \to \mathcal{H}|_ V$ of $\mathcal{O}_ V$-modules.

Proof. You can either prove this by redoing the proof of Lemma 32.20.1 using Lemma 32.10.2 or you can deduce it from Lemma 32.20.1 using the equivalence between quasi-coherent modules and “vector bundles” from Constructions, Section 27.6. We omit the details. $\square$

Comments (0)

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Your email address will not be published.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 0F21. Beware of the difference between the letter 'O' and the digit '0'.