Singular Manual: sres

  | ### 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
      ==>
See betti; fres; hres; ideal; int; lres; minres; module; mres; res; syz.

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