Lemma 64.12.4. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $Y$ is reduced. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(|Y|) \subset |Z|$.

Proof. Assume $f(|Y|) \subset |Z|$. Choose a diagram

$\xymatrix{ V \ar[d]_ b \ar[r]_ h & U \ar[d]^ a \\ Y \ar[r]^ f & X }$

where $U$, $V$ are schemes, and the vertical arrows are surjective and étale. The scheme $V$ is reduced, see Lemma 64.7.1. Hence $h$ factors through $a^{-1}(Z)$ by Schemes, Lemma 26.12.7. So $a \circ h$ factors through $Z$. As $Z \subset X$ is a subsheaf, and $V \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$ we conclude that $X \to Y$ factors through $Z$. $\square$

## Comments (0)

There are also:

• 2 comment(s) on Section 64.12: Reduced spaces

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03JJ. Beware of the difference between the letter 'O' and the digit '0'.