The Stacks Project

Tag 01U1

Lemma 28.22.2. Let $f : X \to S$ be a morphism.

1. If $f$ is locally of finite presentation and generalizations lift along $f$, then $f$ is open.
2. If $f$ is locally of finite presentation and generalizations lift along every base change of $f$, then $f$ is universally open.

Proof. It suffices to prove the first assertion. This reduces to the case where both $X$ and $S$ are affine. In this case the result follows from Algebra, Lemma 10.40.3 and Proposition 10.40.8. $\square$

The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4037–4046 (see updates for more information).

\begin{lemma}
\label{lemma-locally-finite-presentation-universally-open}
Let $f : X \to S$ be a morphism.
\begin{enumerate}
\item If $f$ is locally of finite presentation and generalizations lift
along $f$, then $f$ is open.
\item If $f$ is locally of finite presentation and generalizations lift
along every base change of $f$, then $f$ is universally open.
\end{enumerate}
\end{lemma}

\begin{proof}
It suffices to prove the first assertion.
This reduces to the case where both $X$ and $S$ are affine.
In this case the result follows from
Algebra, Lemma \ref{algebra-lemma-going-up-down-specialization}
and Proposition \ref{algebra-proposition-fppf-open}.
\end{proof}

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