The Stacks project

Lemma 29.21.11. Let $f : X \to Y$ be a morphism of schemes over $S$.

  1. If $X$ is locally of finite presentation over $S$ and $Y$ is locally of finite type over $S$, then $f$ is locally of finite presentation.

  2. If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated and locally of finite type over $S$, then $f$ is of finite presentation.

Proof. Proof of (1). Via Lemma 29.21.2 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is of finite presentation and $A \to B$ is of finite type, then $B \to C$ is of finite presentation. See Algebra, Lemma 10.6.2.

Part (2) follows from (1) and Schemes, Lemmas 26.21.13 and 26.21.14. $\square$

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Your email address will not be published.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 02FV. Beware of the difference between the letter 'O' and the digit '0'.