Lemma 67.16.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then
the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\overline{|f|(|X|)}$, and
for any étale morphism $V \to Y$ the scheme theoretic image of $X \times _ Y V \to V$ is equal to $Z \times _ Y V$.
Proof. Part (1) is true because the reduced induced algebraic space structure on $\overline{|f|(|X|)}$ is the smallest closed subspace of $Y$ through which $f$ factors, see Properties of Spaces, Lemma 66.12.4. Part (2) follows from (1), the fact that $|V| \to |Y|$ is open, and the fact that being reduced is preserved under étale localization. $\square$
Your email address will not be published.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.
Comments (0)