D.4.5.1 reg_CM

Procedure from library mregular.lib (see mregular_lib).

Usage:

reg_CM (i); i ideal

Return:

an integer, the Castelnuovo-Mumford regularity of i-sat.

Assume:

i is a homogeneous ideal of the basering S=K[x(0)..x(n)] where the field K is infinite, and S/i-sat is Cohen-Macaulay. Assume that K[x(n-d),...,x(n)] is a Noether normalization of S/i-sat where d=dim S/i -1. If this is not the case, compute a Noether normalization e.g. by using the proc noetherNormal from algebra.lib.

Note:

The output is reg(X)=reg(i-sat) where X is the arithmetically Cohen-Macaulay subscheme of the projective n-space defined by i. If printlevel > 0 (default = 0) additional information is displayed. In particular, the value of the regularity of the Hilbert function of S/i-sat is given.

Example:

LIB "mregular.lib";
ring s=0,x(0..5),dp;
ideal i=x(2)^2-x(4)*x(5),x(1)*x(2)-x(0)*x(5),x(0)*x(2)-x(1)*x(4),
x(1)^2-x(3)*x(5),x(0)*x(1)-x(2)*x(3),x(0)^2-x(3)*x(4);
reg_CM(i);
→ 2
// Additional information can be obtained as follows:
printlevel = 1;
reg_CM(i);
→ // Ideal i of S defining an arithm. Cohen-Macaulay subscheme X of P5:
→ //   - dimension of X: 2
→ //   - i is saturated: YES
→ //   - regularity of the Hilbert function of S/i-sat: -1
→ //   - time for computing reg(X): 0 sec.
→ // Castelnuovo-Mumford regularity of X:
→ 2