Simple example of a ring which is normal but not CM - MathOverflow

Simple example of a ring which is normal but not CM

2009-10-21T13:42:00Z

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 Cohen-Macaulay. I've found a few in the past, but they were too messy to easily remember and use as test cases. Suggestions?

Answer by Graham Leuschke for Simple example of a ring which is normal but not CM

2009-10-21T14:40:45Z

Segre products of normal CM N-graded K-algebras 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/enescu-faridi.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 a-invariant of A is non-negative.

Answer by David Eisenbud for Simple example of a ring which is normal but not CM

2009-10-23T21:36:27Z

Another family of examples is given by the homogeneous coordinate rings of irregular surfaces (ie 2-dimensional $X$ such that $H^1({\mathcal O}_X) \neq 0$); these surfaces cannot be embedded in any way so that their homogeneous coordinate rings become Cohen-Macaulay. Elliptic scrolls (such as the one in the previous answer) and Abelian surfaces in P4, made from the sections of the Horrocks-Mumford 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 Cohen-Macaulay 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".

Answer by Francisco Perdomo for Simple example of a ring which is normal but not CM

2009-12-08T02:30:13Z

MJ Bertin en Anneaux d'invariantes d'Anneaux de polynômes, en caractéritique p . C.R.Acad.Sci.Paris Sér. A-B 264 1967 A653-A656

"... 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 87-88