The Stacks project

Lemma 10.119.12 (Krull-Akizuki). Let $R$ be a domain with fraction field $K$. Let $L/K$ be a finite extension of fields. Assume $R$ is Noetherian and $\dim (R) = 1$. In this case any ring $A$ with $R \subset A \subset L$ is Noetherian.

Proof. To begin we may assume that $L$ is the fraction field of $A$ by replacing $L$ by the fraction field of $A$ if necessary. Let $I \subset A$ be a nonzero ideal. Clearly $I$ generates $L$ as a $K$-vector space. Hence we see that $I \cap R \not= (0)$. Pick any nonzero $x \in I \cap R$. Then we get $I/xA \subset A/xA$. By Lemma 10.119.11 the $R$-module $A/xA$ has finite length as an $R$-module. Hence $I/xA$ has finite length as an $R$-module. Hence $I$ is finitely generated as an ideal in $A$. $\square$

Comments (3)

Comment #2596 by Aaron Landesman on

Here is a very minor comment: in the proof of Krull-Akizuki, you should assume is nonzero, or else the "Clearly ..." statement is false.

Comment #4515 by awllower on

An alternative way to see that is as follows: Take a non-zero . Since is finite, we can take the minimal polynomial of over . Write it as with . By multiplying by some elements in if necessary, we may assume . Also as is minimal. Now , so .

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