A study of the length function of generalized fractions of modules

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. Let $\underline{x}=x_1,...,x_d$ be a system of parameters of $M$ and $\underline{n}=(n_1,...,n_d)$ a $d$-tuple of positive integers. In this paper we study the length of generalized fractions $M(1/(x_1,...,x_d,1))$ which was introduced by Sharp and Hamieh in \cite{ShH85}. First, we study the growth of the function $J_{\underline{x},M}(\underline{n})=\ell (M(1/(x_1^{n_1},...,x_d^{n_d},1)))-n_1...n_{d}e(\underline{x};M)$. Then we give an explicit calculation for the function $J_{\underline{x},M}(\underline{n})$ in the case where $M$ admits a Macaulayfication. Most previous results on this topic are now easy to understand and to improve.

18 pages