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Tag 0CVT

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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