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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 $A_1={\tt matrix}(M)$.Then sres computes a free resolution of $coker(A_1)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F...
...ver{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow 0.\end{displaymath}

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) $M_i={\tt module} (A_i)$, 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. ( ${\tt L[i]}=M_i$ 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.