Lemma 87.12.2. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 87.12.1. Then $P$ is stable under base change (Formal Spaces, Situation 86.21.6).

Proof. The statement makes sense by Lemma 87.12.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{count}$ such that $B/\mathfrak b \to A/\mathfrak a$ is of finite type where $\mathfrak b \subset B$ and $\mathfrak a \subset A$ are the ideals of topologically nilpotent elements. Since $A$ and $B$ are weakly admissible, the ideals $\mathfrak a$ and $\mathfrak b$ are open. Let $\mathfrak c \subset C$ be the (open) ideal of topologically nilpotent elements. Then we find a surjection $A \widehat{\otimes }_ B C \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ whose kernel is a weak ideal of definition and hence consists of topologically nilpotent elements (please compare with the proof of Formal Spaces, Lemma 86.4.12). Since already $C/\mathfrak c \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ is of finite type as a base change of $B/\mathfrak b \to A/\mathfrak a$ we conclude. $\square$

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