|  |  5.1.73 koszul 
See
 int;
 matrix.Syntax:koszul (int_expression,int_expression)
 koszul (int_expression,ideal_expression)
 koszul (int_expression,int_expression,ideal_expression)Type:matrix
Purpose:koszul(d,n)computes a matrix of the Koszul relations of degree d of
the first n ring variables.
koszul(d,id)computes a matrix of the Koszul relations of degree d of
the generators of the idealid. 
koszul(d,n,id)computes a matrix of the Koszul relations of degree d of
the first n generators of the idealid. 
Note:koszul(1,id), koszul(2,id), ...form a complex, that is, the product
of the matriceskoszul(i,id)andkoszul(i+1,id)equals zero.Example:|  |   ring r=32003,(x,y,z),dp;
  print(koszul(2,3));
==> -y,-z,0, 
==> x, 0, -z,
==> 0, x, y  
  ideal I=xz2+yz2+z3,xyz+y2z+yz2,xy2+y3+y2z;
  print(koszul(1,I));
==> xz2+yz2+z3,xyz+y2z+yz2,xy2+y3+y2z
  print(koszul(2,I));
==> -xyz-y2z-yz2,-xy2-y3-y2z,0,          
==> xz2+yz2+z3,  0,          -xy2-y3-y2z,
==> 0,           xz2+yz2+z3, xyz+y2z+yz2 
  print(koszul(2,I)*koszul(3,I));
==> 0,
==> 0,
==> 0 
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