Lemma 73.9.2. 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.

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

**Proof.**
Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale 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 35.19.1.
$\square$

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## Comments (1)

Comment #1369 by Herman Rohrbach on