The ideal of the 3 by 3 commuting matrices

Given a (homogeneous) ideal , what is the best way to determine whether the ring S/I is Cohen-Macaulay (C.M.)? One way is to compute a minimal finite free resolution of S/I. S/I is C.M. if and only if the length of the resolution is equal to the codimension of S/I. (see for definitions and the basic properties of Cohen- Macaulay rings and modules).

Unfortunately, this method sometimes fails, since it is often too time consuming to compute the finite free resolution. A much easier approach is to use the following well known proposition.

Recall that . In Macaulay, this number can be found from a standard basis by the degree command.

One important problem is the following question of Artin and Hochster. Let be the ideal of the entries of the n by n matrix XY-YX, where , and . Is S/I Cohen-Macaulay? If so, this easily implies that S/I is a normal domain. However, the question has still not been answered for .

In this example, we show that for n = 3, S/I is Cohen-Macaulay. For this example, it is possible but time consuming to compute a finite free resolution, therefore our method of attack is to use the above proposition. The Macaulay session to determine this is given below. The following list of notes refers to this session.

1. The monomial order for the ring S is set to the default: the reverse lexicographic order. This allows the entire computation to take much less time. For example, the session below takes about 30 seconds on either a SUN 3/140 or a Macintosh II.
2. Notice the shorthand ``x[1]-y[9]'' to specify the 18 variables which we need.
3. The matrices X and Y are created using the useful cat command.
4. The random command is used to generate enough random numbers so that we can construct 12 random linear forms.
5. The method we have chosen is not the fastest. For example, the standard basis computation of ``Icut'' is done in the original ring S. However, if we simply reduced I with respect to the random linear forms, we would have an ideal in 6 variables. Computing a standard basis here would be much more efficient. In this example however, more efficiency is not needed.

%  ; first create a ring with 18 variables x[i]'s and y[j]'s.
%  ring S
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  18
!  18 variables, please               ?  x[1]-y[9]
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 16 

%  ; find the ideal I
%  cat x[1] X
! rows ?  0 3 6
! columns ?  0 1 2

%  cat y[1] Y
! rows ?  0 3 6
! columns ?  0 1 2

%  type X
; x[1] x[2] x[3] 
; x[4] x[5] x[6] 
; x[7] x[8] x[9] 

%  type Y
; y[1] y[2] y[3] 
; y[4] y[5] y[6] 
; y[7] y[8] y[9] 

%  mult X Y XY

%  mult Y X YX

%  subtract XY YX XY_YX

%  flatten XY_YX I

%  ; compute a standard basis of I, and its degree.
%  std I I
; 23.4.5.6.7.
; computation complete after degree 7

%  degree I
; codimension : 6
; degree      : 31

%  ; now find a system of parameters for I.
%  ; we need 12 linear forms (since dim(S/I) = 12).
%  ; we choose 12 random linear forms.
%  random 18 12 rand

%  mult S rand linears

%  concat linears I

%  ; compute a standard basis of (I, linears), and its degree.
%  std linears Icut
; 12.3.4.5.6.
; computation complete after degree 6

%  degree Icut
; codimension : 18
; degree      : 31

%  ; both have degree 31, therefore S/I is Cohen-Macaulay.