I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not CohenMacaulay. I've found a few in the past, but they were too messy to easily remember and use as test cases. Suggestions?

Another family of examples is given by the homogeneous coordinate rings of irregular surfaces (ie 2dimensional $X$ such that $H^1({\mathcal O}_X) \neq 0$); these surfaces cannot be embedded in any way so that their homogeneous coordinate rings become CohenMacaulay. Elliptic scrolls (such as the one in the previous answer) and Abelian surfaces in P4, made from the sections of the HorrocksMumford bundle, are such examples. The point is that sufficiently positive, complete embeddings of any smooth variety (or somewhat more generally) will have normal homogeneous coordinate rings, and they will be CohenMacaulay iff the intermediate cohomology of the variety vanishes. All the examples above fall into this category. It's an interesting general question to ask how positive is "sufficiently positive". 


Segre products of normal CM Ngraded Kalgebras A, B are always normal (since they're direct summands of the normal ring A\otimes_K B), but rarely CM. A particular example is here: http://www.math.purdue.edu/~walther/research/segre.ps or also here: http://www.mathstat.dal.ca/~faridi/research/enescufaridi.pdf (Corollary 35) Take A=k[x,y,z]/x^3+y^3+z^3 to be the Fermat cubic, and B=k[a,b]. Then the Segre product A#B is not CM, since the ainvariant of A is nonnegative. 


MJ Bertin en Anneaux d'invariantes d'Anneaux de polynômes, en caractéritique p . C.R.Acad.Sci.Paris Sér. AB 264 1967 A653A656 "... MJ Bertin has made use of Galois descent in order to construct an example of a factorial noetherian ring wich is not Cohen Macaulay..." Robert M. Fossum The example is done completly in The Divisor Class Groups of a Krull Domain, Robert Fossum, Example 16.7 pages 8788 

