Remark 10.160.9. If $k$ is a field then the power series ring $k[[X_1, \ldots , X_ d]]$ is a Noetherian complete local regular ring of dimension $d$. If $\Lambda $ is a Cohen ring then $\Lambda [[X_1, \ldots , X_ d]]$ is a complete local Noetherian regular ring of dimension $d + 1$. Hence the Cohen structure theorem implies that any Noetherian complete local ring is a quotient of a regular local ring. In particular we see that a Noetherian complete local ring is universally catenary, see Lemma 10.105.9 and Lemma 10.106.3.
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