The Stacks project

Lemma 88.13.2. Denote $P$ the property of arrows of $\textit{WAdm}^{Noeth}$ defined in Lemma 88.13.1. Denote $Q$ the property defined in Lemma 88.12.1 viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Denote $R$ the property defined in Lemma 88.11.1 viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Then

  1. $P$ is stable under base change by $Q$ (Formal Spaces, Remark 87.21.10), and

  2. $P + R$ is stable under base change (Formal Spaces, Remark 87.21.9).

Proof. The statement makes sense as each of the properties $P$, $Q$, and $R$ is a local property of morphisms of $\textit{WAdm}^{Noeth}$. Let $\varphi : B \to A$ and $\psi : B \to C$ be morphisms of $\textit{WAdm}^{Noeth}$. If either $Q(\varphi )$ or $Q(\psi )$ then we see that $A \widehat{\otimes }_ B C$ is Noetherian by Formal Spaces, Lemma 87.4.12. Since $R$ implies $Q$ (Lemma 88.12.4), we find that this holds in both cases (1) and (2). This is the first thing we have to check. It remains to show that $C \to A \widehat{\otimes }_ B C$ is flat.

Proof of (1). Fix ideals of definition $I \subset A$ and $J \subset B$. By Lemma 88.12.5 the ring map $B \to C$ is topologically of finite type. Hence $B \to C/J^ n$ is of finite type for all $n \geq 1$. Hence $A \otimes _ B C/J^ n$ is Noetherian as a ring (because it is of finite type over $A$ and $A$ is Noetherian). Thus the $I$-adic completion $A \widehat{\otimes }_ B C/J^ n$ of $A \otimes _ B C/J^ n$ is flat over $C/J^ n$ because $C/J^ n \to A \otimes _ B C/J^ n$ is flat as a base change of $B \to A$ and because $A \otimes _ B C/J^ n \to A \widehat{\otimes }_ B C/J^ n$ is flat by Algebra, Lemma 10.97.2 Observe that $A \widehat{\otimes }_ B C/J^ n = (A \widehat{\otimes }_ B C)/J^ n(A \widehat{\otimes }_ B C)$; details omitted. We conclude that $M = A \widehat{\otimes }_ B C$ is a $C$-module which is complete with respect to the $J$-adic topology such that $M/J^ nM$ is flat over $C/J^ n$ for all $n \geq 1$. This implies that $M$ is flat over $C$ by More on Algebra, Lemma 15.27.4.

Proof of (2). In this case $B \to A$ is adic and hence we have just $A \widehat{\otimes }_ B C = \mathop{\mathrm{lim}}\nolimits A \otimes _ B C/J^ n$. The rings $A \otimes _ B C/J^ n$ are Noetherian by an application of Formal Spaces, Lemma 87.4.12 with $C$ replaced by $C/J^ n$. We conclude in the same manner as before. $\square$

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