The Stacks project

Lemma 66.21.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated of finite type. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent:

  1. the space $|X_ k|$ is finite where $\mathop{\mathrm{Spec}}(k) \to Y$ represents $y$,

  2. $X \to Y$ is quasi-finite at all points of $|X|$ over $y$,

  3. there exists an open subspace $Y' \subset Y$ with $y \in |Y'|$ such that $Y' \times _ Y X \to Y'$ is finite.

If $Y$ is decent these are also equivalent to

  1. the set $f^{-1}(\{ y\} )$ is finite.

Proof. The equivalence of (1) and (2) follows from Lemma 66.18.10 (and the fact that a quasi-separated morphism is decent by Lemma 66.17.2).

Assume the equivalent conditions of (1) and (2). Choose an affine scheme $V$ and an étale morphism $V \to Y$ mapping a point $v \in V$ to $y$. Then $v$ is a generic point of an irreducible component of $V$ by Properties of Spaces, Lemma 64.11.1. Choose an affine scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to V$ is of finite type. The morphism $U \to V$ is quasi-finite at every point lying over $v$ by (2). It follows that the fibre of $U \to V$ over $v$ is finite (Morphisms, Lemma 29.20.14). By Morphisms, Lemma 29.50.1 after shrinking $V$ we may assume that $U \to V$ is finite. Let

\[ R = U \times _{V \times _ Y X} U \]

Since $f$ is quasi-separated, we see that $V \times _ Y X$ is quasi-separated and hence $R$ is a quasi-compact scheme. Moreover the morphisms $R \to V$ is quasi-finite as the composition of an étale morphism $R \to U$ and a finite morphism $U \to V$. Hence we may apply Morphisms, Lemma 29.50.1 once more and after shrinking $V$ we may assume that $R \to V$ is finite as well. This of course implies that the two projections $R \to V$ are finite étale. It follows that $V/R = V \times _ Y X$ is an affine scheme, see Groupoids, Proposition 39.23.9. By Morphisms, Lemma 29.40.8 we conclude that $V \times _ Y X \to V$ is proper and by Morphisms, Lemma 29.43.11 we conclude that $V \times _ Y X \to V$ is finite. Finally, we let $Y' \subset Y$ be the open subspace of $Y$ corresponding to the image of $|V| \to |Y|$. By Morphisms of Spaces, Lemma 65.45.3 we conclude that $Y' \times _ Y X \to Y'$ is finite as the base change to $V$ is finite and as $V \to Y'$ is a surjective étale morphism.

If $Y$ is decent and $f$ is quasi-separated, then we see that $X$ is decent too; use Lemmas 66.17.2 and 66.17.5. Hence Lemma 66.18.10 applies to show that (4) implies (1) and (2). On the other hand, we see that (2) implies (4) by Morphisms of Spaces, Lemma 65.27.9. $\square$


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