### 7.4.3 Syzygies and resolutions (plural)

### Syzygies

Let be a GR-algebra. A left (resp. right) syzygy between elements is a -tuple satisfying

The set of all left (resp. right) syzygies between is a left (resp. right) submodule of .

Remark: With respect to the definitions of `ideal` and `module` (see PLURAL), by `syz` PLURAL understands an inquiry to compute the left syzygy module.

Note, that `rightModulo(M,std(0))` from nctools_lib computes the right syzygy module of `M`.

If `S` is a matrix of a left syzygy module of left submodule given by matrix `M`, then `transpose(S)*transpose(M) = 0` (but, in general, ).

Note, that the syzygy modules of depend on a choice of generators , but one can show that they depend on uniquely up to direct summands.

### Free resolutions

Let and . A free resolution of is a long exact sequence

with

and where the columns of the matrix generate . Note, that resolutions over factor-algebras need not to be of finite length.

### Generalized Hilbert Syzygy Theorem

For a -algebra , generated by variables, there exists a free resolution of length smaller or equal than .

`Example:`

      ring R=0,(x,y,z),dp;
      matrix d[3][3];
      d[1,2]=-z;  d[1,3]=2x;  d[2,3]=-2y;
      def U=nc_algebra(1,d); // this algebra is U(sl_2)
      setring U;
      option(redSB); option(redTail);
      ideal I=x3,y3,z3-z;
      I=std(I);
      I;
      ==> I[1]=z3-z
      ==> I[2]=y3
      ==> I[3]=x3
      ==> I[4]=y2z2-y2z
      ==> I[5]=x2z2+x2z
      ==> I[6]=x2y2z-2xyz2-2xyz+2z2+2z
      resolution resI = mres(I,0);
      resI;
      ==>  1      5      7      3
      ==> U <--  U <--  U <--  U
      ==>
      ==> 0      1      2      3
      ==>
      list l = resI;
      // The matrix A_1 is given by
      print(matrix(l[1]));
      ==> z3-z,y3,x3,y2z2-y2z,x2z2+x2z
      // We see that the columns of A_1 generate I.
      // The matrix A_2 is given by
      print(matrix(l[2]));
      ==> 0,        0,        y2,  x2,  6yz,        -36xy+18z+24,-6xz,
      ==> z2+11z+30,0,        0,   0,   2x2z+12x2,  2x3,         0,
      ==> 0,        z2-11z+30,0,   0,   0,          -2y3,        2y2z-12y2,
      ==> -y,       0,        -z-5,0,   x2y-6xz-30x,9x2,         x3,
      ==> 0,        -x,       0,   -z+5,-y3,        -9y2,        -xy2-4yz+28y
      ideal tst; // now let us show that the resolution is exact
      matrix TST;
      TST = transpose(matrix(l[3]))*transpose(matrix(l[2])); // 2nd term
      size(ideal(TST));
      ==> 0
      TST = transpose(matrix(l[2]))*transpose(matrix(l[1])); // 1st term
      size(ideal(TST));
      ==> 0

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