On the Torsion of Brieskorn Modules of Homogeneous Polynomials

Abstract

-Let f P C[X 1 ; :::; X n ] be a homogeneous polynomial and B( f ) be the corresponding Brieskorn module. We describe the torsion of the Brieskorn module B( f ) for n 2 and show that any torsion element has order 1. For n > 2, we find some examples in which the torsion order is strictly greater than 1. The Milnor algebra and the Brieskorn module.Let f P R C[x 1 ; :::; x n ] be a homogeneous polynomial of degree d > 1.Then the Koszul complex of the partial derivatives f j @f @x j ; j 1; :::; n in R can be identified to the complexwhere V j denotes the regular differential forms of degree j on C n . Let J f be the Jacobian ideal spanned by the partial derivatives f j , j 1; :::; n, in R and M( f ) R=J f be the Milnor algebra of f . One has the following obvious isomorphism of graded vector spacesHere, for any graded C[t]-module M, the shifted module M(m) is defined by setting M(m) s M ms for all s P Z:We define the (algebraic) Brieskorn module as the quotient