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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.

  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}

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