On the last Hilbert–Samuel coefficient of isolated singularities

J. Lipman proved that the last Hilbert–Samuel coefficient of normal ideals of a 2-dimensional complete local ring R   are bounded by the geometric genus of X=Spec(R)X=Spec(R). In this paper we extend this result to ideals I of a d-dimensional Cohen–Macaulay local ring R such that the associated graded ring of R   with respect to InIn is Cohen–Macaulay for n≫0n≫0. We study the ideals such that their last Hilbert–Samuel coefficients equal the geometric genus.