Singular

5.1.145 sres

Syntax:

sres ( ideal_expression, int_expression )
sres ( module_expression, int_expression )

Type:

resolution

Purpose:

computes a free resolution of an ideal or module with Schreyer's method. The ideal, resp. module, has to be a standard basis. More precisely, let M be given by a standard basis and .Then sres computes a free resolution of

If the int expression k is not zero then the computation stops after k steps and returns a list of modules (given by standard bases) , i=1..k.
sres(M,0) returns a list of n modules where n is the number of variables of the basering.

Even if sres does not compute a minimal resolution, the betti command gives the true betti numbers! In many cases of interest sres is much faster than any other known method. Let list L=sres(M,0); then L[1]=M is identical to the input, L[2] is a standard basis with respect to the Schreyer ordering of the first syzygy module of L[1], etc. ( in the notations from above.)

Note:

Accessing single elements of a resolution may require some partial computations to be finished and may therefore take some time.

Example:

ring r=31991,(t,x,y,z,w),ls; ideal M=t2x2+tx2y+x2yz,t2y2+ty2z+y2zw, t2z2+tz2w+xz2w,t2w2+txw2+xyw2; M=std(M); resolution L=sres(M,0); L; ==> 1 35 141 209 141 43 4 ==> r <-- r <-- r <-- r <-- r <-- r <-- r ==> ==> 0 1 2 3 4 5 6 ==> resolution not minimized yet ==> print(betti(L),"betti"); ==> 0 1 2 3 4 5 ==> ------------------------------------------ ==> 0: 1 - - - - - ==> 1: - - - - - - ==> 2: - - - - - - ==> 3: - 4 - - - - ==> 4: - - - - - - ==> 5: - - - - - - ==> 6: - - 6 - - - ==> 7: - - 9 16 2 - ==> 8: - - - 2 5 1 ==> ------------------------------------------ ==> total: 1 4 15 18 7 1 ==>