## C.3 Syzygies and resolutions

### Syzygies

Let be a quotient of and let be a submodule of . Then the module of syzygies (or 1st syzygy module, module of relations) of , syz(), is defined to be the kernel of the map .

The k-th syzygy module is defined inductively to be the module of syzygies of the -stsyzygy module.

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.

`Example:`

        ring R= 0,(u,v,x,y,z),dp;
        ideal i=ux, vx, uy, vy;
        print(syz(i));
      ==> -y,0, -v,0,
      ==> 0, -y,u, 0,
      ==> x, 0, 0, -v,
      ==> 0, x, 0, u

### Free resolutions

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

where the columns of the matrix generate . Note that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states that for there exists a ("minimal") resolution of length not exceeding the number of variables.

`Example:`

        ring R= 0,(u,v,x,y,z),dp;
        ideal I = ux, vx, uy, vy;
        resolution resI = mres(I,0); resI;
      ==>  1      4      4      1
      ==> R <--  R <--  R <--  R
      ==>
      ==> 0      1      2      3
      ==>
        // The matrix A_1 is given by
        print(matrix(resI[1]));
      ==> vy,uy,vx,ux
        // We see that the columns of A_1 generate I.
        // The matrix A_2 is given by
        print(matrix(resI[3]));
      ==> u,
      ==> -v,
      ==> -x,
      ==> y

### Betti numbers and regularity

Let be a graded ring (e.g., ) and let be a graded submodule. Let

be a minimal free resolution of considered with homogeneous maps of degree 0. Then the graded Betti number of is the minimal number of generators in degree of the -th syzygy module of (i.e., the -st syzygy module of ). Note that, by definition, the -th syzygy module of is and the 1st syzygy module of is .

The regularity of is the smallest integer such that

`Example:`

        ring R= 0,(u,v,x,y,z),dp;
        ideal I = ux, vx, uy, vy;
        resolution resI = mres(I,0); resI;
      ==>  1      4      4      1
      ==> R <--  R <--  R <--  R
      ==>
      ==> 0      1      2      3
      ==>
        // the betti number:
        print(betti(resI), "betti");
      ==>            0     1     2     3
      ==> ------------------------------
      ==>     0:     1     -     -     -
      ==>     1:     -     4     4     1
      ==> ------------------------------
      ==> total:     1     4     4     1
      ==>
        // the regularity:
        regularity(resI);
      ==> 2

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