  | ## C.3 Syzygies and resolutions ### Syzygies Let 53#53 be a quotient of 659#659 and let 660#660 be a submodule of 624#624. Then the module of syzygies (or 1st syzygy module, module of relations) of 253#253, syz(253#253), is defined to be the kernel of the map 661#661. The k-th syzygy module is defined inductively to be the module of syzygies of the 662#662-stsyzygy module. Note, that the syzygy modules of 253#253 depend on a choice of generators 663#663. But one can show that they depend on 253#253 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 664#664 and 665#665. A free resolution of 13#13 is a long exact sequence 666#666 where the columns of the matrix 166#166generate 253#253. Note that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states that for 623#623there 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 53#53 be a graded ring (e.g., 623#623) and let 667#667 be a graded submodule. Let 668#668 be a minimal free resolution of 669#669 considered with homogeneous maps of degree 0. Then the graded Betti number 670#670 of 669#669 is the minimal number of generators 671#671 in degree 172#172 of the 55#55-th syzygy module of 669#669 (i.e., the 672#672-st syzygy module of 253#253). Note that, by definition, the 2#2-th syzygy module of 669#669 is 624#624 and the 1st syzygy module of 669#669 is 253#253. The regularity of 253#253 is the smallest integer 178#178 such that 673#673 `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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