Singular

2.3.6 Resolution

There are several commands in SINGULAR for computing free resolutions. The most general command is res(... ,n) which determines heuristically what method to use for the given problem. It computes the free resolution up to the length , where corresponds to the full resolution.

Here we use the possibility to inspect the calculation process using the option prot.

ring R;      // the default ring in char 32003
R;
==> //   characteristic : 32003
==> //   number of vars : 3
==> //        block   1 : ordering dp
==> //                  : names    x y z
==> //        block   2 : ordering C
ideal I = x4+x3y+x2yz,x2y2+xy2z+y2z2,x2z2+2xz3,2x2z2+xyz2;
option(prot);
resolution rs = res(I,0);
==> using lres
==> 4(m0)4(m1).5(m1)g.g6(m1)...6(m2)..

Disable this protocol with

option(noprot);

When we enter the name of the calculated resolution, we get a pictorial description of the minimized resolution where the exponents denote the rank of the free modules. Note that the calculated resolution itself may not yet be minimal.

rs;
==> 1      4      5      2      0
==>R  <-- R  <-- R  <-- R  <-- R
==>
==>0      1      2      3      4
print(betti(rs),"betti");
==>            0     1     2     3
==> ------------------------------
==>     0:     1     -     -     -
==>     1:     -     -     -     -
==>     2:     -     -     -     -
==>     3:     -     4     1     -
==>     4:     -     -     1     -
==>     5:     -     -     3     2
==> ------------------------------
==> total:     1     4     5     2

In order to minimize the resolution, that is to calculate the maps of the minimal free resolution, we use the command minres:

rs=minres(rs);

A single module in this resolution is obtained (as usual) with the brackets [ and ]. The print command can be used to display a module in a more readable format:

print(rs[3]);
==> z3,   -xyz-y2z-4xz2+16z3,
==> 0,    -y2,
==> -y+4z,48z,
==> x+2z, 48z,
==> 0,    x+y-z

In this case, the output is to be interpreted as follows: the 3rd syzygy module of R/I, rs[3], is the rank-2-submodule of generated by the vectors and .