The Stacks project

Lemma 74.9.4. Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular.

Proof. By Lemma 74.9.3 we know that $Y$ is locally Noetherian. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. It suffices to prove that the local rings of $V$ are all regular local rings, see Properties, Lemma 28.9.2. Choose a scheme $U$ and a surjective ├ętale morphism $U \to X \times _ Y V$. As $f$ is surjective and flat the morphism of schemes $U \to V$ is surjective and flat. By assumption $U$ is a regular scheme in particular all of its local rings are regular (by the lemma above). Hence the lemma follows from Algebra, Lemma 10.110.9. $\square$

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