The Stacks project

Lemma 86.11.3. Consider the property $P$ on arrows of $\textit{WAdm}^{adic*}$ defined in Lemma 86.11.1. Then $P$ is stable under base change as defined in Formal Spaces, Remark 85.17.8.

Proof. The statement makes sense by Lemma 86.11.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{adic*}$ and that as a topological $B$-algebra we have $A = B\{ x_1, \ldots , x_ r\} /J$ for some closed ideal $J$. Then $A \widehat{\otimes }_ B C$ is isomorphic to the quotient of $C\{ x_1, \ldots , x_ r\} /J'$ where $J'$ is the closure of $JC\{ x_1, \ldots , x_ r\} $. Some details omitted. $\square$

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