The Stacks project

Lemma 31.12.6. Let $X$ be an integral scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then

  1. $\mathcal{F}$ has finite rank $r \geq 0$, and

  2. there exists a nonempty open $U \subset X$ such that $\mathcal{F}|_ U$ is free of rank $r$.

Proof. See More on Algebra, Lemma 15.23.6 for the algebraic version. $\square$

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