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Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.

We assume that $j_*\mathcal{E}$ is a vector bundle. In particular, it is Zariski locally trivial. Can we express a Zariski cover that trivializes $j_{*}\mathcal{E}$ in terms of a Zariski cover that trivializes $\mathcal{E}$?

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  • $\begingroup$ It seems strange that $j_*\mathcal{E}$ would be a coherent $\mathcal{O}_X$-module. Of course, this is quasi-coherent, but $j$ is not proper (unless $X \setminus U$ is empty...) so coherent sheaves on $U$ don't go to coherent sheaves on $X$. Do you mean there is some vector bundle $\mathcal{E}'$ on $X$ that restricts to $\mathcal{E}$ on $U$? (e.g. as in Hartshorne Ex. II.5.15?) $\endgroup$
    – Eric Canton
    Commented Jun 28, 2019 at 19:27
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    $\begingroup$ As the open is of codimension two and X normal, the pushforward is always reflexive and coherent. I make the extra assumption that it is a vector bundle. $\endgroup$
    – prochet
    Commented Jun 28, 2019 at 20:37
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    $\begingroup$ I doubt that this question has a sensible answer. $\endgroup$
    – Angelo
    Commented Jun 29, 2019 at 12:27

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