The Stacks project

(Locally) quasi-finite morphisms are stable under base change.

Lemma 29.20.13. Let $f : X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Denote $f' : X' \to S'$ the base change of $f$ by $g$ and denote $g' : X' \to X$ the projection. Assume $X$ is locally of finite type over $S$.

  1. Let $U \subset X$ (resp. $U' \subset X'$) be the set of points where $f$ (resp. $f'$) is quasi-finite. Then $U' = U \times _ S S' = (g')^{-1}(U)$.

  2. The base change of a locally quasi-finite morphism is locally quasi-finite.

  3. The base change of a quasi-finite morphism is quasi-finite.

Proof. The first and second assertion follow from the corresponding algebra result, see Algebra, Lemma 10.122.8 (combined with the fact that $f'$ is also locally of finite type by Lemma 29.15.4). By the above, Lemma 29.20.9 and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 26.19.3 we see that the base change of a quasi-finite morphism is quasi-finite. $\square$

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