The Stacks project

Lemma 66.27.2. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be morphisms of algebraic spaces over $S$. Denote $f' : X' \to Y'$ the base change of $f$ by $g$. Denote $g' : X' \to X$ the projection. Assume $f$ is locally of finite type. Let $W \subset |X|$, resp. $W' \subset |X'|$ be the set of points where $f$, resp. $f'$ is quasi-finite.

  1. $W \subset |X|$ and $W' \subset |X'|$ are open,

  2. $W' = (g')^{-1}(W)$, i.e., formation of the locus where $f$ is quasi-finite commutes with base change,

  3. the base change of a locally quasi-finite morphism is locally quasi-finite, and

  4. the base change of a quasi-finite morphism is quasi-finite.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a scheme $V'$ and a surjective étale morphism $V' \to Y' \times _ Y V$. Set $U' = V' \times _ V U$ so that $U' \to X'$ is a surjective étale morphism as well. Picture

\[ \vcenter { \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V } } \quad \text{lying over}\quad \vcenter { \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } } \]

Choose $u \in |U|$ with image $x \in |X|$. The property of being "locally quasi-finite" is étale local on the source-and-target, see Descent, Remark 35.32.7. Hence Lemmas 66.22.5 and 66.22.7 apply and we see that $f : X \to Y$ is quasi-finite at $x$ if and only if $U \to V$ is quasi-finite at $u$. Similarly for $f' : X' \to Y'$ and the morphism $U' \to V'$. Hence parts (1), (2), and (3) reduce to Morphisms, Lemmas 29.20.13 and 29.55.2. Part (4) follows from (3) and Lemma 66.8.4. $\square$

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