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Tag 01PC

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

Lemma 27.16.2. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_X$-modules $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation if and only if $M$ is an $R$-module of finite presentation.

Proof. Assume $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation. By Lemma 27.16.1 we see that $M$ is a finite $R$-module. Choose a surjection $R^n \to M$ with kernel $K$. By Schemes, Lemma 25.5.4 there is a short exact sequence $$ 0 \to \widetilde{K} \to \bigoplus \mathcal{O}_X^{\oplus n} \to \widetilde{M} \to 0 $$ By Modules, Lemma 17.11.3 we see that $\widetilde{K}$ is a finite type $\mathcal{O}_X$-module. Hence by Lemma 27.16.1 again we see that $K$ is a finite $R$-module. Hence $M$ is an $R$-module of finite presentation. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 1891–1896 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-presentation-module}
    Let $X = \Spec(R)$ be an affine scheme. The quasi-coherent sheaf
    of $\mathcal{O}_X$-modules $\widetilde M$ is an $\mathcal{O}_X$-module of
    finite presentation if and only if $M$ is an $R$-module of finite presentation.
    \end{lemma}
    
    \begin{proof}
    Assume $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation.
    By Lemma \ref{lemma-finite-type-module} we see that $M$ is a finite $R$-module.
    Choose a surjection $R^n \to M$ with kernel $K$. By
    Schemes, Lemma \ref{schemes-lemma-spec-sheaves}
    there is a short exact sequence
    $$
    0 \to \widetilde{K} \to
    \bigoplus \mathcal{O}_X^{\oplus n} \to
    \widetilde{M} \to 0
    $$
    By
    Modules, Lemma
    \ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}
    we see that $\widetilde{K}$ is a finite type $\mathcal{O}_X$-module.
    Hence by Lemma \ref{lemma-finite-type-module}
    again we see that $K$ is a finite $R$-module.
    Hence $M$ is an $R$-module of finite presentation.
    \end{proof}

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