On the last Hilbert-Samuel coefficient of isolated singularities

TL;DR

This paper investigates the behavior of the last Hilbert-Samuel coefficient in the context of isolated singularities, extending Lipman's results on the finiteness of certain invariants in algebraic geometry.

Contribution

It extends Lipman's finiteness result for the Hilbert-Samuel coefficient to Cohen-Macaulay rings with Cohen-Macaulay associated graded rings for large powers.

Findings

Established conditions for the finiteness of the last Hilbert-Samuel coefficient.

Extended Lipman's results to higher-dimensional Cohen-Macaulay rings.

Provided new insights into the structure of singularities via Hilbert coefficients.

Abstract

In 1978 Lipman presented a proof of the existence of a desingularization for any excellent surface. The strategy of Lipman's proof is based on the finiteness of the number H(R) defined as the supreme of the second Hilbert-Samuel coefficient I, where I range the set of normal m-primary ideals of a Noetherian complete local ring (R,m). The problem studied in the paper is the extension of the result of Lipman on H(R) to m-primary ideals I of a d-dimensional Cohen-Macaulay ring R such that the associated graded ring of R with respect to I^n is Cohen-Macaulay for n>> 0.