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Tag 05NZ

27.19. Flat modules

On any ringed space $(X, \mathcal{O}_X)$ we know what it means for an $\mathcal{O}_X$-module to be flat (at a point), see Modules, Definition 17.16.1 (Definition 17.16.3). For quasi-coherent sheaves on an affine scheme this matches the notion defined in the algebra chapter.

Lemma 27.19.1. Let $X = \mathop{\rm Spec}(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a flat $\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module.

Proof. Flatness of $\mathcal{F}$ may be checked on the stalks, see Modules, Lemma 17.16.2. The same is true in the case of modules over a ring, see Algebra, Lemma 10.38.19. And since $\mathcal{F}_x = M_{\mathfrak p}$ if $x$ corresponds to $\mathfrak p$ the lemma is true. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 2233–2268 (see updates for more information).

    \section{Flat modules}
    \label{section-flat}
    
    \noindent
    On any ringed space $(X, \mathcal{O}_X)$
    we know what it means for an $\mathcal{O}_X$-module
    to be flat (at a point), see
    Modules, Definition \ref{modules-definition-flat}
    (Definition \ref{modules-definition-flat-at-point}).
    For quasi-coherent sheaves on an affine scheme this matches the notion
    defined in the algebra chapter.
    
    \begin{lemma}
    \label{lemma-flat-module}
    \begin{slogan}
    Flatness is the same for modules and sheaves.
    \end{slogan}
    Let $X = \Spec(R)$ be an affine scheme.
    Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$.
    The quasi-coherent sheaf $\mathcal{F}$ is a flat
    $\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module.
    \end{lemma}
    
    \begin{proof}
    Flatness of $\mathcal{F}$ may be checked on the stalks, see
    Modules, Lemma \ref{modules-lemma-flat-stalks-flat}.
    The same is true in the case of modules over a ring, see
    Algebra, Lemma \ref{algebra-lemma-flat-localization}.
    And since $\mathcal{F}_x = M_{\mathfrak p}$ if $x$ corresponds
    to $\mathfrak p$ the lemma is true.
    \end{proof}

    Comments (2)

    Comment #2345 by Jonathan Gruner on January 8, 2017 a 3:41 pm UTC

    In Lemma 27.19.1, there seems to be a typo: “of if and only if”.

    Also, in the text above the lemma, one could add that this is a statement about quasi-coherent sheaves: “For quasi-coherent sheaves on an affine scheme, this matches the notion defined in the algebra chapter.”

    Comment #2414 by Johan (site) on February 17, 2017 a 1:43 pm UTC

    Thanks. Fixed here.

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