Lemma 88.14.3. Let \varphi : A \to B in \textit{WAdm}^{Noeth}. Denote \mathfrak a \subset A and \mathfrak b \subset B the ideals of topologically nilpotent elements. Assume A/\mathfrak a \to B/\mathfrak b is of finite type. Let \mathfrak q \subset B be rig-closed. The residue field \kappa of the local ring B/\mathfrak q is a finite type A/\mathfrak a-algebra.
Proof. Let \mathfrak q \subset \mathfrak m \subset B be the unique maximal ideal containing \mathfrak q. Then \mathfrak b \subset \mathfrak m. Hence A/\mathfrak a \to B/\mathfrak b \to B/\mathfrak m = \kappa is of finite type. \square
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