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5.1.119 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 A1 = matrix(M). Then sres computes a free resolution of coker(A1) = F0∕M

...−→  F2−A2→ F1 A−1→ F0 −→ F0∕M  −→ 0.
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) Mi = module(Ai), 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. (L[i] = Mi in the notations from above.)

Note:

Accessing single elements of a resolution may require that some partial computations have 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

See betti; hres; ideal; int; lres; minres; module; mres; res; syz.


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