Lemma 87.14.5. 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 87.14.3 (combined with Lemma 87.12.4) 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 87.14.4. $\square$

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