The Stacks project

Proof. Let $R$ be a complete local Noetherian ring. Let $\mathfrak p \subset R$ be a prime. Then $R/\mathfrak p$ is also a complete local Noetherian ring, see Lemma 10.160.2. Hence it suffices to show that a Noetherian complete local domain $R$ is N-2. By Lemmas 10.161.5 and 10.160.11 we reduce to the case $R = k[[X_1, \ldots , X_ d]]$ where $k$ is a field or $R = \Lambda [[X_1, \ldots , X_ d]]$ where $\Lambda $ is a Cohen ring.

In the case $k[[X_1, \ldots , X_ d]]$ we reduce to the statement that a field is N-2 by Lemma 10.161.17. This is clear. In the case $\Lambda [[X_1, \ldots , X_ d]]$ we reduce to the statement that a Cohen ring $\Lambda $ is N-2. Applying Lemma 10.161.16 once more with $x = p \in \Lambda $ we reduce yet again to the case of a field. Thus we win. $\square$

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