### 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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