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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?

David

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