Lemma 38.19.6. Let A \to B be a local ring map of local rings which is essentially of finite type. Let N be a finite B-module which is flat as an A-module. If A is a valuation ring, then any element of N has a content ideal I \subset A (More on Algebra, Definition 15.24.1). Also, I is a principal ideal.
Proof. The final statement follows from the fact that I is a finitely generated ideal by More on Algebra, Lemma 15.24.2 and Algebra, Lemma 10.50.15.
Proof of existence of I. Let A \subset A^ h be the henselization. Let B' be the localization of B \otimes _ A A^ h at the maximal ideal \mathfrak m_ B \otimes A^ h + B \otimes \mathfrak m_{A^ h}. Then B \to B' is flat, hence faithfully flat. Let N' = N \otimes _ B B'. Let x \in N and let x' \in N' be the image. We claim that for an ideal I \subset A we have x \in IN \Leftrightarrow x' \in IN'. Namely, N/IN \to N'/IN' is the tensor product of B \to B' with N/IN and B \to B' is universally injective by Algebra, Lemma 10.82.11. By More on Algebra, Lemma 15.124.6 and Algebra, Lemma 10.50.17 the map A \to A^ h defines an inclusion preserving bijection I \mapsto IA^ h on sets of ideals. We conclude that x has a content ideal in A if and only if x' has a content ideal in A^ h. The assertion for x' \in N' follows from Lemma 38.19.5 and Algebra, Lemma 10.89.6. \square
Comments (2)
Comment #8087 by Laurent Moret-Bailly on
Comment #8204 by Stacks Project on