Lemma 20.49.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$. The category of $\mathcal{O}_ X$-modules which are summands of finite free $\mathcal{O}_ X$-modules is equivalent to the category of finite projective $R$-modules.

Proof. Observe that a finite projective $R$-module is the same thing as a summand of a finite free $R$-module. The equivalence is given by the functor $\mathcal{E} \mapsto \Gamma (X, \mathcal{E})$. The inverse functor is given by the construction of Modules, Lemma 17.10.5. $\square$

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