# Locus of trivialization of an extension of a vector bundle

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 $$\mathcal{E}$$?

• 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?) – Eric Canton Jun 28 at 19:27
• 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. – prochet Jun 28 at 20:37
• I doubt that this question has a sensible answer. – Angelo Jun 29 at 12:27