Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 29.44.14. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms.

  1. If $g \circ f$ is finite and $g$ separated then $f$ is finite.

  2. If $g \circ f$ is integral and $g$ separated then $f$ is integral.

Proof. Assume $g \circ f$ is finite (resp. integral) and $g$ separated. The base change $X \times _ Z Y \to Y$ is finite (resp. integral) by Lemma 29.44.6. The morphism $X \to X \times _ Z Y$ is a closed immersion as $Y \to Z$ is separated, see Schemes, Lemma 26.21.11. A closed immersion is finite (resp. integral), see Lemma 29.44.12. The composition of finite (resp. integral) morphisms is finite (resp. integral), see Lemma 29.44.5. Thus we win. $\square$

Comments (0)

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Your email address will not be published.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 035D. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.44.14 as pdf