The Stacks project

Proof. First, note that if $M$ is a projective $A$-module and $A \to B$ is a ring map, then $M \otimes _ A B$ is a projective $B$-module, see Algebra, Lemma 10.94.1. Hence if $U$ is an affine open such that $\mathcal{F}(U)$ is a projective $\mathcal{O}_ X(U)$-module, then the standard open $D(f)$ is an affine open such that $\mathcal{F}(D(f))$ is a projective $\mathcal{O}_ X(D(f))$-module for all $f \in \mathcal{O}_ X(U)$. Assume (2) holds. Let $U \subset X$ be an arbitrary affine open. We can find an open covering $U = \bigcup _{j = 1, \ldots , m} D(f_ j)$ by finitely many standard opens $D(f_ j)$ such that for each $j$ the open $D(f_ j)$ is a standard open of some $U_ i$, see Schemes, Lemma 26.11.5. Hence, if we set $A = \mathcal{O}_ X(U)$ and if $M$ is an $A$-module such that $\mathcal{F}|_ U$ corresponds to $M$, then we see that $M_{f_ j}$ is a projective $A_{f_ j}$-module. It follows that $A \to B = \prod A_{f_ j}$ is a faithfully flat ring map such that $M \otimes _ A B$ is a projective $B$-module. Hence $M$ is projective by Algebra, Theorem 10.95.6. $\square$

Proof. See Algebra, Lemma 10.94.1. $\square$