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29.24 Submersive morphisms

Definition 29.24.1. 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 29.24.2. 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 29.24.3. 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.

Comments (2)

Comment #4577 by Andy on

I'm just curious, why does this section exist? It is a property of schemes with no algebraic/other criterion to show when a morphism is submersive/universally submersive. Is this property important? Is it hard to verify? Is this section just to give this property a name?

I hope that didn't sound rude, I'm genuinely just curious.

Comment #4578 by Andy on

I would like to rephrase my previous comment in a more constructive way. I also am interested in this property of morphisms. Does it have any equivalent conditions? Does it have any important consequences?

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