|  |  7.4.3 Syzygies and resolutions (plural) 
  Syzygies Let be a GR-algebra. A left (resp. right) syzygy
between  elements  is
a  -tuple  satisfying 
 The set of all left (resp. right) syzygies between
  is a left (resp. right) submodule  of  . 
Remark:
With respect to the definitions of
 idealandmodule(see  PLURAL),
bysyzPLURAL understands an inquiry to compute the left syzygy module. 
Note, that rightModulo(M,std(0))from  nctools_lib computes the right syzygy module ofM. 
If Sis a matrix of a left syzygy module of left submodule given by matrixM,
thentranspose(S)*transpose(M) = 0(but, in general, ). 
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. 
  Free resolutions Let and  .
A free resolution of  is a long exact sequence 
 
 with
  
and where the columns of the matrix
 generate  . Note, that resolutions over factor-algebras need not to be of finite length. 
  Generalized Hilbert Syzygy Theorem For a -algebra  , generated by  variables, there exists a free resolution of length smaller or equal than  . 
 
Example:|  | ring R=0,(x,y,z),dp;
matrix d[3][3];
d[1,2]=-z;  d[1,3]=2x;  d[2,3]=-2y;
def U=nc_algebra(1,d); // this algebra is U(sl_2)
setring U;
option(redSB); option(redTail);
ideal I=x3,y3,z3-z;
I=std(I);
I;
==> I[1]=z3-z
==> I[2]=y3
==> I[3]=x3
==> I[4]=y2z2-y2z
==> I[5]=x2z2+x2z
==> I[6]=x2y2z-2xyz2-2xyz+2z2+2z
resolution resI = mres(I,0);
resI;
==>  1      5      7      3      
==> U <--  U <--  U <--  U
==> 
==> 0      1      2      3      
==> 
list l = resI;
// The matrix A_1 is given by
print(matrix(l[1]));
==> z3-z,y3,x3,y2z2-y2z,x2z2+x2z
// We see that the columns of A_1 generate I.
// The matrix A_2 is given by
print(matrix(l[2]));
==> 0,        0,        y2,  x2,  6yz,        -36xy+18z+24,-6xz,       
==> z2+11z+30,0,        0,   0,   2x2z+12x2,  2x3,         0,          
==> 0,        z2-11z+30,0,   0,   0,          -2y3,        2y2z-12y2,  
==> -y,       0,        -z-5,0,   x2y-6xz-30x,9x2,         x3,         
==> 0,        -x,       0,   -z+5,-y3,        -9y2,        -xy2-4yz+28y
ideal tst; // now let us show that the resolution is exact
matrix TST;
TST = transpose(matrix(l[3]))*transpose(matrix(l[2])); // 2nd term
size(ideal(TST));
==> 0
TST = transpose(matrix(l[2]))*transpose(matrix(l[1])); // 1st term
size(ideal(TST));
==> 0
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