Lemma 17.14.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. If all stalks $\mathcal{O}_{X, x}$ are local rings, then any direct summand of a finite locally free $\mathcal{O}_ X$-module is finite locally free.
Proof. Assume $\mathcal{F}$ is a direct summand of the finite locally free $\mathcal{O}_ X$-module $\mathcal{H}$. Let $x \in X$ be a point. Then $\mathcal{H}_ x$ is a finite free $\mathcal{O}_{X, x}$-module. Because $\mathcal{O}_{X, x}$ is local, we see that $\mathcal{F}_ x \cong \mathcal{O}_{X, x}^{\oplus r}$ for some $r$, see Algebra, Lemma 10.78.2. By Lemma 17.11.6 we see that $\mathcal{F}$ is free of rank $r$ in an open neighbourhood of $x$. (Note that $\mathcal{F}$ is of finite presentation as a summand of $\mathcal{H}$.) $\square$
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