# The Stacks Project

## 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 2274–2309 (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}

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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