The Stacks project

Lemma 67.37.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. There is a maximal open subspace $U \subset X$ such that $f|_ U : U \to Y$ is smooth. Moreover, formation of this open commutes with base change by

  1. morphisms which are flat and locally of finite presentation,

  2. flat morphisms provided $f$ is locally of finite presentation.

Proof. The existence of $U$ follows from the fact that the property of being smooth is Zariski (and even étale) local on the source, see Lemma 67.37.4. Moreover, this lemma allows us to translate properties (1) and (2) into the case of morphisms of schemes. The case of schemes is Morphisms, Lemma 29.34.15. Some details omitted. $\square$

Comments (2)

Comment #11277 by Torsten Wedhorn on

One could make the statement slightly more precise by saying that there exists a unique maximal open subspace such that ... Similarly, for the analogous statement for algebraic stacks (Tag 0DZR).

Comment #11278 by Laurent Moret-Bailly on

I agree with Torsten, except I would write "largest open subspace" instead of "unique maximal open subspace".

There are also:

  • 2 comment(s) on Section 67.37: Smooth morphisms

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