The Stacks project

Lemma 88.23.1. In the situation above. If $f$ is locally of finite type, then $f_{/T}$ is locally of finite type.

Proof. (Finite type morphisms of formal algebraic spaces are discussed in Formal Spaces, Section 87.24.) Namely, suppose that $Z \to X$ is a morphism from a scheme into $X$ such that $|Z|$ maps into $T$. From the cartesian square above we see that $Z \times _ X X'$ is an algebraic space representing $Z \times _{X_{/T}} X'_{/T'}$. Since $Z \times _ X X' \to Z$ is locally of finite type by Morphisms of Spaces, Lemma 67.23.3 we conclude. $\square$

Comments (3)

Comment #1965 by Brian Conrad on

In the statement of the result, "algebraic spaces" should be "formal algebraic spaces". On the 2nd line of the proof remove "". On the 3rd line of the proof, in the first fiber product the placement of and should be swapped.

Comment #1966 by Brian Conrad on

Please disregard the first sentence of my preceding comment.

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Your email address will not be published.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 0AQ9. Beware of the difference between the letter 'O' and the digit '0'.