Simple example of a ring which is normal but not CM - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:19:50Z http://mathoverflow.net/feeds/question/1652 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1652/simple-example-of-a-ring-which-is-normal-but-not-cm Simple example of a ring which is normal but not CM David Speyer 2009-10-21T13:42:00Z 2009-12-08T02:30:13Z <p>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?</p> http://mathoverflow.net/questions/1652/simple-example-of-a-ring-which-is-normal-but-not-cm/1668#1668 Answer by Graham Leuschke for Simple example of a ring which is normal but not CM Graham Leuschke 2009-10-21T14:40:45Z 2009-10-21T14:40:45Z <p>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. </p> <p>A particular example is here: </p> <p><a href="http://www.math.purdue.edu/~walther/research/segre.ps" rel="nofollow">http://www.math.purdue.edu/~walther/research/segre.ps</a> </p> <p>or also here: </p> <p><a href="http://www.mathstat.dal.ca/~faridi/research/enescu-faridi.pdf" rel="nofollow">http://www.mathstat.dal.ca/~faridi/research/enescu-faridi.pdf</a> (Corollary 35) </p> <p>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.</p> http://mathoverflow.net/questions/1652/simple-example-of-a-ring-which-is-normal-but-not-cm/2194#2194 Answer by David Eisenbud for Simple example of a ring which is normal but not CM David Eisenbud 2009-10-23T21:36:27Z 2009-10-23T21:36:27Z <p>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. </p> <p>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".</p> http://mathoverflow.net/questions/1652/simple-example-of-a-ring-which-is-normal-but-not-cm/8156#8156 Answer by Francisco Perdomo for Simple example of a ring which is normal but not CM Francisco Perdomo 2009-12-08T02:30:13Z 2009-12-08T02:30:13Z <p><i>MJ Bertin en Anneaux d'invariantes d'Anneaux de polynômes, en caractéritique p </i>. C.R.Acad.Sci.Paris Sér. A-B 264 1967 A653-A656</p> <p>"... 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</p> <p>The example is done completly in <i>The Divisor Class Groups of a Krull Domain</i>, Robert Fossum, Example 16.7 pages 87-88</p>