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
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
Comments (0)