Hilbert Functions and Pseudo-Rational Local Rings of Dimension Two

Abstract

Let Q be an analytically unramified Cohen-Macaulay local ring of dimension 2, with maximal ideal m and infinite residue field k. If a is an m-primary ideal of Q, a∗ will denote its integral closure, λ(a) = l(Q/a∗), e(a) will denote its multiplicity, θ(a) = 2. λ(a)−e(a) and θ ¯ (a) = lim θ(an)/n. This paper uses explicit formulae for θ(ar), θ(arbs) related to earlier results of Narita to prove that θ ¯ (ab) = θ ¯ (a) + θ ¯ (b), and that the identity θ(ab) = θ(a) + θ(b) holding for all m-primary ideals a, b characterises the pseudo-rational local rings of J. Lipman among normal Q. This is used to prove a number of results concerning the normal genus of an ideal and pseudo-rational local rings. In the last section, an expression for θ(a) is obtained in the case where Q is regular which is related to the theory of infinitely near points of D. G. Northcott.