Lemma 88.17.4. Consider the properties $P(\varphi )=$“$\varphi $ is rig-smooth” and $Q(\varphi )$=“$\varphi $ is adic” on arrows of $\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark 87.21.10.
Proof. The statement makes sense by Lemma 88.17.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ and that $B \to A$ is rig-smooth and $B \to C$ is adic (Formal Spaces, Definition 87.6.1). Then we can choose an ideal of definition $I \subset B$ such that the topology on $A$ and $C$ is the $I$-adic topology. In this situation it follows immediately that $A \widehat{\otimes }_ B C$ is rig-smooth over $(C, IC)$ by Lemma 88.4.5. $\square$
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