5.1.85 nres

Syntax:

nres ( ideal_expression, int_expression )
nres ( module_expression, int_expression )

Type:

resolution

Purpose:

computes a free resolution of an ideal or module M which is minimized from the second module on (by the standard basis method).

More precisely, let =matrix(M), then nres computes a free resolution of

where the columns of the matrix are the given set of generators of M. If the int expression k is not zero then the computation stops after k steps and returns a list of modules , i=1..k.
nres(M,0) returns a list of n modules where n is the number of variables of the basering. Let list L=nres(M,0); then L[1]=M is identical to the input, L[2] is a minimal set of generators for the first syzygy module of L[1], etc. ( in the notations from above).

Example:

ring r=31991,(t,x,y,z,w),ls; ideal M=t2x2+tx2y+x2yz,t2y2+ty2z+y2zw, t2z2+tz2w+xz2w,t2w2+txw2+xyw2; resolution L=nres(M,0); L; → 1 4 15 18 7 1 → r <-- r <-- r <-- r <-- r <-- r → → 0 1 2 3 4 5 → resolution not minimized yet →

See hres; ideal; lres; module; mres; res; resolution; sres.