  | ### 5.1.73 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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