### 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.

User manual for Singular version 4-0-3, 2016, generated by texi2html.