The Stacks project

Lemma 31.12.2. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $r$ be a cardinal. The following are equivalent

  1. $\mathcal{F}$ has rank $r$,

  2. for all $U \subset X$ nonempty affine open $\mathcal{F}(U)$ is an $\mathcal{O}_ X(U)$-module of rank $r$, and

  3. for some $U \subset X$ nonempty affine open $\mathcal{F}(U)$ is an $\mathcal{O}_ X(U)$-module of rank $r$.

Proof. Omitted. $\square$

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