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

Lemma 28.24.10. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ locally finite presentation, $\mathcal{F}$ of finite type, $X = \text{Supp}(\mathcal{F})$, and $\mathcal{F}$ flat over $Y$. Then $f$ is universally open.

Proof. By Lemmas 28.24.6, 28.20.4, and 28.5.3 the assumptions are preserved under base change. By Lemma 28.22.2 it suffices to show that generalizations lift along $f$. This follows from Algebra, Lemma 10.40.12. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4360–4367 (see updates for more information).

    \begin{lemma}
    \label{lemma-pf-flat-module-open}
    Let $f : X \to Y$ be a morphism of schemes.
    Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
    Assume $f$ locally finite presentation, $\mathcal{F}$ of
    finite type, $X = \text{Supp}(\mathcal{F})$, and
    $\mathcal{F}$ flat over $Y$. Then $f$ is universally open.
    \end{lemma}
    
    \begin{proof}
    By Lemmas \ref{lemma-base-change-module-flat},
    \ref{lemma-base-change-finite-presentation}, and
    \ref{lemma-support-finite-type}
    the assumptions are preserved under base change.
    By Lemma \ref{lemma-locally-finite-presentation-universally-open}
    it suffices to show that generalizations lift along $f$.
    This follows from Algebra, Lemma \ref{algebra-lemma-going-down-flat-module}.
    \end{proof}

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