Torsion of differentials of hypersurfaces with isolated singularities

Abstract

Let R = K[ X 1, X 2,…, X N ], where K is an algebraically closed field of characteristic 0 and consider the reduced, affine hypersurface algebra with an isolated singularity A = R (F) , where F ϵ K[ X 1, X 2,…, X N ]. For such algebras A the torsion (sub) modules of (Kaehler) differentials T(Ω A K N − 1) and Ω A K N are finite dimensional. Unlike in the case of a quasi-homogeneous hypersurface T(Ω A K N − 1) is not always cyclic even if some permutation of ∂F ∂X 1 ,…, ∂F ∂X N is an R-sequence. The main result of this paper proves that for reduced hypersurfaces with only isolated singularities dim KT(Ω A K N − 1) = dim K Ω A K N . We give an example of a reduced plane curve with a single isolated singularity at the origin such that the partial derivatives of F do not form an R-sequence.