
Lemma 10.155.17. Let $R$ be a ring. If $R$ is Noetherian, a domain, and N-2, then so is $R[[x]]$.

Proof. Observe that $R[[x]]$ is Noetherian by Lemma 10.30.2. Let $R' \supset R$ be the integral closure of $R$ in its fraction field. Because $R$ is N-2 this is finite over $R$. Hence $R'[[x]]$ is finite over $R[[x]]$. By Lemma 10.36.9 we see that $R'[[x]]$ is a normal domain. Apply Lemma 10.155.16 to the element $x \in R'[[x]]$ to see that $R'[[x]]$ is N-2. Then Lemma 10.155.7 shows that $R[[x]]$ is N-2. $\square$

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