TORSION OF DIFFERENTIALS OF HYPERSURFACES WITH ISOLATED SINGULARITIES

Let R = K[X(1), X(2),...,X(N)], where K is an algebraically closed held of characteristic 0 and consider the reduced, affine hypersurface algebra with an isolated singularity A = R/(F), where F is an element of K[X(1), X(2),...,X(N). For such algebras A the torsion (sub) modules of(Kaehler) differentials T(Omega(A/K)(N-1)) and Omega(A/K)(N) are finite dimensional. Unlike in the case of a quasi-homogeneous hypersurface T(Omega(A/K)(N-1)) is not always cyclic even if some permutation of partial derivative F/partial derivative X(1),...,partial derivative F/partial derivative X(N) is an R-sequence. The main result of this paper proves that for reduced hypersurfaces with only isolated singularities dim(K) T (Omega(A/K)(N-1)) = dim(K) Omega(A/K)(N-1). 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.