The Stacks project

Lemma 66.37.9. 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 ├ętale) local on the source, see Lemma 66.37.4. 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. $\square$

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