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. $\square$


Comments (1)

Comment #836 by on

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


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01TM. Beware of the difference between the letter 'O' and the digit '0'.