## Tag `00NW`

Chapter 10: Commutative Algebra > Section 10.77: Finite projective modules

Definition 10.77.1. Let $R$ be a ring and $M$ an $R$-module.

- We say that $M$ is
locally freeif 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 freeif 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 $R_{f_i}^{\oplus r}$.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 18121–18134 (see updates for more information).

```
\begin{definition}
\label{definition-locally-free}
Let $R$ be a ring and $M$ an $R$-module.
\begin{enumerate}
\item We say that $M$ is {\it locally free} if we can cover $\Spec(R)$ by
standard opens $D(f_i)$, $i \in I$ such that $M_{f_i}$ is a free
$R_{f_i}$-module for all $i \in I$.
\item We say that $M$ is {\it finite locally free} if we can choose
the covering such that each $M_{f_i}$ is finite free.
\item We say that $M$ is {\it finite locally free of rank $r$}
if we can choose the covering such that each $M_{f_i}$ is isomorphic
to $R_{f_i}^{\oplus r}$.
\end{enumerate}
\end{definition}
```

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