The Stacks project

35.16 Variants on descending properties

Sometimes one can descend properties, which are not local. We put results of this kind in this section. See also Section 35.11 on descending properties of morphisms, such as smoothness.

Lemma 35.16.1. If $f : X \to Y$ is a flat and surjective morphism of schemes and $X$ is reduced, then $Y$ is reduced.

Proof. The result follows by looking at local rings (Schemes, Definition 26.12.1) and Algebra, Lemma 10.164.2. $\square$

Lemma 35.16.2. 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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