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Let $X$ be a scheme, and suppose $\mathcal{F}$ is a locally free sheaf on $X$. Suppose there exist sheaves of modules $\mathcal{G}, \mathcal{H}$ on $X$ such that $\mathcal{F} \cong \mathcal{G} \oplus \mathcal{H}$. Suppose $\mathcal{G}$ is locally free. Does this imply that $\mathcal{H}$ is locally free?


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Yes, and we do not need to assume that $\mathcal{G}$ is locally free (this happens automatically). The question is local, so we reduce immediately to the case that $X = \text{Spec } A$ is affine and $\mathcal{F}$ corresponds to a projective $A$-module. A direct summand of a projective module is certainly projective, since a module is projective if and only if it is a direct summand of a free module.

Edit: As Zhen Lin points out, there is something slightly nontrivial happening here. Namely, in order for this argument to work, we must assume that $\mathcal{F}$ is coherent. The fact that I am using is that a finitely presented module is projective if and only if it is locally free.

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Isn't there some non-trivial fact about the relation between projective modules and locally free modules you're using here? – Zhen Lin May 25 '12 at 18:36

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