Lemma 87.14.3. 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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