Lemma 87.14.4. 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. Let $\mathfrak p = \varphi ^{-1}(\mathfrak q) \subset A$. Let $\mathfrak a \subset A$ be the ideal of topologically nilpotent elements. The following are equivalent

the residue field $\kappa $ of $B/\mathfrak q$ is finite over $A/\mathfrak a$,

$\mathfrak p \subset A$ is rig-closed,

$A/\mathfrak p \subset B/\mathfrak q$ is a finite extension of rings.

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