The Stacks project

Lemma 66.12.4. 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 ├ętale. The scheme $V$ is reduced, see Lemma 66.7.1. 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 $Z$. $\square$

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