The Stacks project

Lemma 70.8.5. Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Assume $Y$ is integral. Then there exists an integral closed subspace $X' \subset X$ such that $f' = f|_{X'} : X' \to Y$ is an alteration.

Proof. Let $V \subset Y$ be a nonempty open affine (Decent Spaces, Theorem 66.10.2). Let $\eta \in V$ be the generic point. Then $X_\eta $ is a nonempty proper algebraic space over $\eta $. Choose a closed point $x \in |X_\eta |$ (exists because $|X_\eta |$ is a quasi-compact, sober topological space, see Decent Spaces, Proposition 66.12.4 and Topology, Lemma 5.12.8.) Let $X'$ be the reduced induced closed subspace structure on $\overline{\{ x\} } \subset |X|$ (Properties of Spaces, Definition 64.12.5. Then $f' : X' \to Y$ is surjective as the image contains $\eta $. Also $f'$ is proper as a composition of a closed immersion and a proper morphism. Finally, the fibre $X'_\eta $ has a single point; to see this use Decent Spaces, Lemma 66.18.6 for both $X \to Y$ and $X' \to Y$ and the point $\eta $. Since $Y$ is decent and $X' \to Y$ is separated we see that $X'$ is decent (Decent Spaces, Lemmas 66.17.2 and 66.17.5). Thus $f'$ is an alteration by Lemma 70.8.4. $\square$


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