Singular

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