|  |  C.3 Syzygies and resolutions 
  Syzygies  Let be a quotient of ![$\hbox{Loc}_< K[\underline{x}]$](sing_620.png) 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 ![$R = \hbox{Loc}_< K[\underline{x}]$](sing_584.png) 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., ![$R = \hbox{Loc}_< K[\underline{x}]$](sing_584.png) ) 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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