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Tag 01PB

Chapter 27: Properties of Schemes > Section 27.16: Characterizing modules of finite type and finite presentation

Lemma 27.16.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is a finite type $\mathcal{O}_X$-module if and only if $M$ is a finite $R$-module.

Proof. Assume $\widetilde M$ is a finite type $\mathcal{O}_X$-module. This means there exists an open covering of $X$ such that $\widetilde M$ restricted to the members of this covering is globally generated by finitely many sections. Thus there also exists a standard open covering $X = \bigcup_{i = 1, \ldots, n} D(f_i)$ such that $\widetilde M|_{D(f_i)}$ is generated by finitely many sections. Thus $M_{f_i}$ is finitely generated for each $i$. Hence we conclude by Algebra, Lemma 10.23.2. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 1871–1877 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-type-module}
    Let $X = \Spec(R)$ be an affine scheme.
    The quasi-coherent sheaf of $\mathcal{O}_X$-modules
    $\widetilde M$ is a finite type $\mathcal{O}_X$-module
    if and only if $M$ is a finite $R$-module.
    \end{lemma}
    
    \begin{proof}
    Assume $\widetilde M$ is a finite type $\mathcal{O}_X$-module.
    This means there exists an open covering of $X$ such that
    $\widetilde M$ restricted to the members of this covering is
    globally generated by finitely many sections.
    Thus there also exists a standard open covering
    $X = \bigcup_{i = 1, \ldots, n} D(f_i)$ such that $\widetilde M|_{D(f_i)}$
    is generated by finitely many sections. Thus $M_{f_i}$ is finitely
    generated for each $i$. Hence we conclude by
    Algebra, Lemma \ref{algebra-lemma-cover}.
    \end{proof}

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