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Tag 01UA

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

Lemma 28.24.9. A flat morphism locally of finite presentation is universally open.

Proof. This follows from Lemmas 28.24.8 and Lemma 28.22.2 above. We can also argue directly as follows.

Let $f : X \to S$ be flat locally of finite presentation. To show $f$ is open it suffices to show that we may cover $X$ by open affines $X = \bigcup U_i$ such that $U_i \to S$ is open. By definition we may cover $X$ by affine opens $U_i \subset X$ such that each $U_i$ maps into an affine open $V_i \subset S$ and such that the induced ring map $\mathcal{O}_S(V_i) \to \mathcal{O}_X(U_i)$ is of finite presentation. Thus $U_i \to V_i$ is open by Algebra, Proposition 10.40.8. The lemma follows. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4330–4333 (see updates for more information).

    \begin{lemma}
    \label{lemma-fppf-open}
    A flat morphism locally of finite presentation is universally open.
    \end{lemma}
    
    \begin{proof}
    This follows from Lemmas \ref{lemma-generalizations-lift-flat} and
    Lemma \ref{lemma-locally-finite-presentation-universally-open} above.
    We can also argue directly as follows.
    
    \medskip\noindent
    Let $f : X \to S$ be flat locally of finite presentation.
    To show $f$ is open it suffices to show that we may cover
    $X$ by open affines $X = \bigcup U_i$ such that $U_i \to S$
    is open. By definition we may cover $X$ by
    affine opens $U_i \subset X$ such that each $U_i$ maps
    into an affine open $V_i \subset S$ and such that
    the induced ring map $\mathcal{O}_S(V_i) \to \mathcal{O}_X(U_i)$ is
    of finite presentation. Thus $U_i \to V_i$ is open by
    Algebra, Proposition \ref{algebra-proposition-fppf-open}.
    The lemma follows.
    \end{proof}

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