= Locally free sheaf
= Locally free
{synonym}
= Locally free coherent sheaf
{synonym}
A sheaf of $\mathcal O_X$-modules is locally free of finite rank when every point has an open neighborhood $U$ on which it is isomorphic to $\mathcal O_U^{\oplus r}$ for some finite $r$. If $X$ is connected, the rank is constant.
A sheaf of modules is locally free of rank $r$ if each point has a neighborhood $U$ on which it is isomorphic to $\mathcal O_U^r$. Finite-rank locally free sheaves correspond to <vector bundles> in the standard geometric settings.
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