The Dual Hilbert–Samuel Function of a Maximal Cohen-Macaulay Module

Abstract

Let R be a Cohen–Macaulay local ring with a canonical module ω R . Let I be an 𝔪-primary ideal of R and M, a maximal Cohen–Macaulay R-module. We call the function n⟼ℓ(Hom R (M, ω R /I n+1ω R )) the dual Hilbert–Samuel function of M with respect to I . By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.