The Stacks project

Lemma 65.23.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and $Y$ is locally Noetherian, then $X$ is locally Noetherian.

Proof. Let

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

be a commutative diagram where $U$, $V$ are schemes and the vertical arrows are surjective ├ętale. If $f$ is locally of finite type, then $U \to V$ is locally of finite type. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. By Morphisms, Lemma 29.15.6 we see that $U$ is locally Noetherian, which means that $X$ is locally Noetherian. $\square$


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Your email address will not be published.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 04ZK. Beware of the difference between the letter 'O' and the digit '0'.