Lemma 88.14.5. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak q \subset B$ be rig-closed. If $A/I$ is Jacobson for some ideal of definition $I \subset A$, then $\mathfrak p = \varphi ^{-1}(\mathfrak q) \subset A$ is rig-closed.

**Proof.**
By Lemma 88.14.3 (combined with Lemma 88.12.4) the residue field $\kappa $ of $B/\mathfrak q$ is of finite type over $A/\mathfrak a$. Since $A/\mathfrak a$ is Jacobson, we see that $\kappa $ is finite over $A/\mathfrak a$ by Algebra, Lemma 10.35.18. We conclude by Lemma 88.14.4.
$\square$

## 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Your email address will not be published.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.

## Comments (0)