|
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.
|