### 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 .

User manual for Singular version 4.4.1, 2025, generated by texi2html.