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5.1.68 koszul

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 ideal id.

koszul(d,n,id) computes a matrix of the Koszul relations of degree d of the first n generators of the ideal id.

Note:
koszul(1,id), koszul(2,id), ... form a complex, that is, the product of the matrices koszul(i,id) and koszul(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 
See int; matrix.

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