 Locally free sheaf

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.