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Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact.

Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $E$ is a finitely generated locally free sheaf of $\mathcal{O}_X$-modules on $U$.

$\textbf{My question}$ is: can we always extend $E$ to get a vector bundle on $X$? I believe the statement is true but I cannot find any reference on it.

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No, this is false. Take $X=\mathbb{R}^3$ and $U=\mathbb{R}^3\smallsetminus\{0\} $, with $\mathcal{O}_X$ the sheaf of complex $C^{\infty}$ functions. Line bundles on $U$ are parametrized by $H^1(U, \mathcal{O}_U^*)$, which is isomorphic to $H^2(U,\mathbb{Z})=\mathbb{Z}$ by the exponential exact sequence. Similarly $H^1(X, \mathcal{O}_X^*)\cong H^2(X,\mathbb{Z})=0$, so there are nontrivial line bundles on $U$ which do not extend to $X$.

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    $\begingroup$ It is easy to write down such a vector bundle. Take the tangent bundle of the sphere and pull it back via $x \mapsto x/|x|$. $\endgroup$
    – Ben McKay
    Commented Apr 5, 2016 at 18:38

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