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. $\square$

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