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Let $X$ be a variety with $Y \subsetneq X$ a proper closed subvariety. Let $Z$ denote the blow-up of $X$ along $Y$. Let $f: Z \rightarrow X$ be the canonical map. Suppose that we have a coherent sheaf of $\mathcal{O}_X$-modules $F$ such that $F|_{X-Y}$ is locally free and $f^*F$ is locally free on $Z$. Under what conditions will a submodule $G \subset f^*F$ be the pullback $f^*(G_0)$ of a free submodule $G_0 \subset F$? For example, the fibers of $G$ should be constant of $f^{-1}(Y)$ but I don't think that is sufficient. I'd be happy with just an explanation of the case where $X$ is affine space and $Y$ is a point.

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