Torsion of differentials of hypersurfaces with isolated singularities

AbstractLet R = K[X1,X2,…,XN], 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[X1,X2,…,XN]. For such algebras A the torsion (sub) modules of (Kaehler) differentials T(ΩAKN − 1) and ΩAKN are finite dimensional. Unlike in the case of a quasi-homogeneous hypersurface T(ΩAKN − 1) is not always cyclic even if some permutation of ∂F∂X1,…,∂F∂XN is an R-sequence. The main result of this paper proves that for reduced hypersurfaces with only isolated singularities dimKT(ΩAKN − 1) = dimK ΩAKN. 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