  | ## C.3 Syzygies and resolutions ### Syzygies Let 53#53 be a quotient of 657#657 and let 658#658 be a submodule of 622#622. Then the module of syzygies (or 1st syzygy module, module of relations) of 251#251, syz(251#251), is defined to be the kernel of the map 659#659. The k-th syzygy module is defined inductively to be the module of syzygies of the 660#660-stsyzygy module. Note, that the syzygy modules of 251#251 depend on a choice of generators 661#661. But one can show that they depend on 251#251 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 662#662 and 663#663. A free resolution of 13#13 is a long exact sequence 664#664 where the columns of the matrix 166#166generate 251#251. Note that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states that for 621#621there 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., 621#621) and let 665#665 be a graded submodule. Let 666#666 be a minimal free resolution of 667#667 considered with homogeneous maps of degree 0. Then the graded Betti number 668#668 of 667#667 is the minimal number of generators 669#669 in degree 172#172 of the 55#55-th syzygy module of 667#667 (i.e., the 670#670-st syzygy module of 251#251). Note that, by definition, the 2#2-th syzygy module of 667#667 is 622#622 and the 1st syzygy module of 667#667 is 251#251. The regularity of 251#251 is the smallest integer 177#177 such that 671#671 `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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