# Quotient of a Regular local ring.

Is the quotient of a regular local ring by a prime ideal Cohen-Macaulay? If so, how can we see this, if not, is there a counterexample?

We know that a regular local ring is a UFD, so $0$ is a prime ideal, so in this case, the quotient is a regular local ring and hence a CM ring.

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If the prime ideal is generated by a subset of a system of parameters, then the quotient by such a prime ideal is a regular local ring and hence a CM. – messi Jun 20 '12 at 11:31

Any complete local noetherian ring is a quotient of a regular local ring. This is called Cohen's structure theorem for noetherian complete local ring (see for instance Matsumura Commutative Ring Theory Theorem 29.4). So any complete noetherian integral domain is a quotient of a local regular ring by a prime ideal. To construct a counterexample to your question, it is thus enough to exhibit a complete local noetherian domain which is not Cohen-Macaulay. The ring $k[X^4,X^3Y,XY^3,Y^4]$ (which is a non-CM domain) is usually a good starting point.