On Adjusted Hilbert-Samuel Functions

Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring and M a finitely generated R-module of dimension d. Let $\mathfrak {q}$ be a parameter ideal of M. Consider an adjusted Hilbert-Samuel function in n defined by

$ f_{\mathfrak {q},M}(n)=\ell (M/\mathfrak {q}^{n+1}M)-\sum \limits _{i=0}^{d}\text {adeg}_{i}(\mathfrak {q};M) \left (\begin {array}{c} n+i \\ i \end {array}\right ), $

where $\text {adeg}_{i}(\mathfrak {q};M)$ is the ith arithmetic degree of M with respect to $\mathfrak {q}$. In this paper, we prove that if $\mathfrak {q}$ is a distinguished parameter ideal then there exists an integer n ₀ such that $f_{\mathfrak {q}, M}(n)\geq 0$ for all n≥n ₀. Moreover, if M is sequentially generalized Cohen-Macaulay, then n ₀ exists independently of the choice of $\frak q$.