# The Stacks Project

## Tag 00NW

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

1. 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 $i \in I$.
2. We say that $M$ is finite locally free if we can choose the covering such that each $M_{f_i}$ is finite free.
3. 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}

There are no comments yet for this tag.

There are also 2 comments on Section 10.77: Commutative Algebra.

## Add a comment on tag 00NW

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).