# The Stacks Project

## Tag: 03ER

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Definition 17.28.1. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings.
1. A presheaf $\mathcal{F}$ of $\mathcal{O}$-modules is called flat if the functor $$\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F}$$ is exact.
2. A map $\mathcal{O} \to \mathcal{O}'$ of presheaves of rings is called flat if $\mathcal{O}'$ is flat as a presheaf of $\mathcal{O}$-modules.
3. If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings and $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then we say $\mathcal{F}$ is flat if the functor $$\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_\mathcal{O} \mathcal{F}$$ is exact.
4. A map $\mathcal{O} \to \mathcal{O}'$ of sheaves of rings on a site is called flat if $\mathcal{O}'$ is flat as a sheaf of $\mathcal{O}$-modules.

\begin{definition}
\label{definition-flat}
Let $\mathcal{C}$ be a category.
Let $\mathcal{O}$ be a presheaf of rings.
\begin{enumerate}
\item A presheaf $\mathcal{F}$ of $\mathcal{O}$-modules is called
{\it flat} if the functor
$$\textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_{p, \mathcal{O}} \mathcal{F}$$
is exact.
\item A map $\mathcal{O} \to \mathcal{O}'$ of presheaves of rings
is called {\it flat} if $\mathcal{O}'$ is flat as a presheaf of
$\mathcal{O}$-modules.
\item If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings
and $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then we
say $\mathcal{F}$ is {\it flat} if the functor
$$\textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{G} \mapsto \mathcal{G} \otimes_\mathcal{O} \mathcal{F}$$
is exact.
\item A map $\mathcal{O} \to \mathcal{O}'$ of sheaves of rings on a site
is called {\it flat} if $\mathcal{O}'$ is flat as a sheaf of
$\mathcal{O}$-modules.
\end{enumerate}
\end{definition}


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