Definition 10.78.1. Let $R$ be a ring and $M$ an $R$-module.

We say that $M$ is

*locally free*if we can cover $\mathop{\mathrm{Spec}}(R)$ by standard opens $D(f_ i)$, $i \in I$ such that $M_{f_ i}$ is a free $R_{f_ i}$-module for all $i \in I$.We say that $M$ is

*finite locally free*if we can choose the covering such that each $M_{f_ i}$ is finite free.We say that $M$ is

*finite locally free of rank $r$*if we can choose the covering such that each $M_{f_ i}$ is isomorphic to $R_{f_ i}^{\oplus r}$.

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