## Tag `01U3`

Chapter 28: Morphisms of Schemes > Section 28.24: Flat morphisms

Definition 28.24.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules.

- We say $f$ is
flat at a point $x \in X$if the local ring $\mathcal{O}_{X, x}$ is flat over the local ring $\mathcal{O}_{S, f(x)}$.- We say that $\mathcal{F}$ is
flat over $S$ at a point $x \in X$if the stalk $\mathcal{F}_x$ is a flat $\mathcal{O}_{S, f(x)}$-module.- We say $f$ is
flatif $f$ is flat at every point of $X$.- We say that $\mathcal{F}$ is
flat over $S$if $\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.

The code snippet corresponding to this tag is a part of the file `morphisms.tex` and is located in lines 4197–4211 (see updates for more information).

```
\begin{definition}
\label{definition-flat}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules.
\begin{enumerate}
\item We say $f$ is {\it flat at a point $x \in X$} if the
local ring $\mathcal{O}_{X, x}$ is flat over the local ring
$\mathcal{O}_{S, f(x)}$.
\item We say that $\mathcal{F}$ is {\it flat over $S$ at a point $x \in X$}
if the stalk $\mathcal{F}_x$ is a flat $\mathcal{O}_{S, f(x)}$-module.
\item We say $f$ is {\it flat} if $f$ is flat at every point of $X$.
\item We say that $\mathcal{F}$ is {\it flat over $S$} if
$\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.
\end{enumerate}
\end{definition}
```

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