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Tag 040G

28.23. Submersive morphisms

Definition 28.23.1. Let $f : X \to Y$ be a morphism of schemes.

  1. We say $f$ is submersive1 if the continuous map of underlying topological spaces is submersive, see Topology, Definition 5.6.3.
  2. We say $f$ is universally submersive if for every morphism of schemes $Y' \to Y$ the base change $Y' \times_Y X \to Y'$ is submersive.

We note that a submersive morphism is in particular surjective.

Lemma 28.23.2. The base change of a universally submersive morphism of schemes by any morphism of schemes is universally submersive.

Proof. This is immediate from the definition. $\square$

Lemma 28.23.3. The composition of a pair of (universally) submersive morphisms of schemes is (universally) submersive.

Proof. Omitted. $\square$

  1. This is very different from the notion of a submersion of differential manifolds.

The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4124–4172 (see updates for more information).

\section{Submersive morphisms}
\label{section-submersive}

\begin{definition}
\label{definition-submersive}
Let $f : X \to Y$ be a morphism of schemes.
\begin{enumerate}
\item We say $f$ is {\it submersive}\footnote{This is very different
from the notion of a submersion of differential manifolds.}
if the continuous map of underlying topological spaces is submersive, see
Topology, Definition \ref{topology-definition-submersive}.
\item We say $f$ is {\it universally submersive} if for every
morphism of schemes $Y' \to Y$ the base change
$Y' \times_Y X \to Y'$ is submersive.
\end{enumerate}
\end{definition}

\noindent
We note that a submersive morphism is in particular surjective.

\begin{lemma}
\label{lemma-base-change-universally-submersive}
The base change of a universally submersive morphism of schemes
by any morphism of schemes is universally submersive.
\end{lemma}

\begin{proof}
This is immediate from the definition.
\end{proof}

\begin{lemma}
\label{lemma-composition-universally-submersive}
The composition of a pair of (universally) submersive morphisms of
schemes is (universally) submersive.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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