Lemma 35.19.2. 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.**
This lemma reduces to the following algebra statement: If $A \to B$ is a faithfully flat, finitely presented ring homomorphism with $B$ Noetherian and regular, then $A$ is Noetherian and regular. We see that $A$ is Noetherian by Algebra, Lemma 10.164.1 and regular by Algebra, Lemma 10.110.9.
$\square$

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