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Tag 07T9

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

Lemma 28.24.13. Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.

Proof. Using the characterization of scheme theoretic support on affines as given in Lemma 28.5.4 we reduce to Algebra, Lemma 10.39.4. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4469–4475 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-pullback-support}
    Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be
    a finite type quasi-coherent $\mathcal{O}_X$-module with scheme
    theoretic support $Z \subset X$. If $f$ is flat,
    then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    Using the characterization of scheme theoretic support on affines
    as given in Lemma \ref{lemma-scheme-theoretic-support} we reduce to
    Algebra, Lemma \ref{algebra-lemma-annihilator-flat-base-change}.
    \end{proof}

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