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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.

  1. 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 $\mathcal{O}_{S, f(x)}$.
  2. 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.
  3. We say $f$ is flat if $f$ is flat at every point of $X$.
  4. 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 4200–4214 (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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