The Stacks project

Remark 85.17.10 (Another variant for Noetherian). Let $P$ and $Q$ be local properties of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 85.17.5. We say $P$ is stable under base change by $Q$ if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ satisfying $P(B \to A)$ and $Q(B \to C)$, then $A \widehat{\otimes }_ B C$ is adic Noetherian and $P(C \to A \widehat{\otimes }_ B C)$ holds. Arguing exactly as in the proof of Lemma 85.17.7 we obtain the following statement: given morphisms $f : X \to Y$ and $g : Y \to Z$ of locally Noetherian formal algebraic spaces over $S$ such that

  1. the equivalent conditions of Lemma 85.17.3 hold for $f$ and $P$,

  2. the equivalent conditions of Lemma 85.17.3 hold for $g$ and $Q$,

then the equivalent conditions of Lemma 85.17.3 hold for $\text{pr}_2 : X \times _ Y Z \to Z$ and $P$.


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