Question 1 Could you find a non Cohen-Macaulay ring $A$ without zero divisors.

I would like $A$ to be as simple as possible. For instance, I want $A$ to be finitely generated alegbra over $\mathbb{C}$.

Comment I found plenty of examples of non Cohen-Macaulay rings in wiki and from other sources. But those rings are not integral domains.

Question 2 A projective curve is called ACM if it's homogeneous coordinate ring is Cohen-Macaulay. It would be great if one give me an example of non ACM curve in $\mathbb{P}^3$.

up vote 3 down vote accepted
  1. Since any one-dimensional integral domain is CM, let's look a little further. Then we find the two-dimensional ring $R=K[X^4,X^3Y,XY^3,Y^4]$ which is not CM.

  2. $R=K[X,Y,Z,W]/(X,W)\cap(Y,Z)$ (the disjoint union of two lines in $\mathbb P^3$) is not CM.

(The example given in 1. is also good as it is the coordinate ring of the rational quartic curve in $\mathbb P^3$, that is, $K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$.)

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