The Stacks project

A flat and surjective morphism of algebraic spaces with a reduced source has a reduced target.

Lemma 73.8.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced.

Proof. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. 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. In this way we reduce the problem to the case of schemes (as reducedness of $X$ and $Y$ is defined in terms of reducedness of $U$ and $V$, see Properties of Spaces, Section 65.7). The case of schemes is Descent, Lemma 35.18.1. $\square$


Comments (1)

Comment #1369 by Herman Rohrbach on

Suggested slogan: A flat and surjective morphism of algebraic spaces with a reduced source has a reduced target.


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