Singular Manual: Syzygies and resolutions

  | ## C.3 Syzygies and resolutions ### Syzygies The k-th syzygy module is defined inductively to be the module of syzygies of the syzygy module. `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 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 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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