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
$A_1={\tt matrix}(M)$.
Then sres
computes a free resolution of
$coker(A_1)=F_0/M$
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow 0.$$
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)
$M_i={\tt module}(A_i)$, 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 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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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
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