The Stacks project

Remark 87.21.9 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 87.21.5. We say $P$ is stable under base change if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ the property $P(B \to A)$ implies both that $A \widehat{\otimes }_ B C$ is adic Noetherian¹ and that $P(C \to A \widehat{\otimes }_ B C)$. In exactly the same way we obtain a variant of Lemma 87.21.7 for morphisms between locally Noetherian formal algebraic spaces over $S$.