Lemma 73.8.5. Let $f : X \to Y$ be a smooth morphism of algebraic spaces. If $Y$ is reduced, then $X$ is reduced. If $f$ is surjective and $X$ is reduced, then $Y$ is reduced.
Proof. Choose a commutative diagram
where $U$ and $V$ are schemes, the vertical arrows are surjective and étale, and $U \to X \times _ Y V$ is surjective étale. Observe that $X$ is a reduced algebraic space if and only if $U$ is a reduced scheme by our definition of reduced algebraic spaces in Properties of Spaces, Section 65.7. Similarly for $Y$ and $V$. The morphism $U \to V$ is a smooth morphism of schemes, see Morphisms of Spaces, Lemma 66.37.4. Since being reduced is local for the smooth topology for schemes (Descent, Lemma 35.17.1) we see that $U$ is reduced if $V$ is reduced. On the other hand, if $X \to Y$ is surjective, then $U \to V$ is surjective and in this case if $U$ is reduced, then $V$ is reduced. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.