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 $S$.

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

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

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

## Comments (1)

Comment #836 by Johan Commelin on