A study of the length function of generalized fractions of modules

TL;DR

This paper investigates the growth of a length function associated with generalized fractions of modules over Noetherian local rings, providing explicit calculations in cases where modules admit a Macaulayfication.

Contribution

It introduces a detailed study of the length function of generalized fractions, offering explicit formulas and insights especially for modules with Macaulayfication, enhancing understanding of prior results.

Findings

Analyzed the growth of the length function $J_{\underline{x}, M}(\underline{n})$.

Provided explicit calculations for modules with Macaulayfication.

Simplified understanding of previous results on generalized fractions.

Abstract

Let $(R, \frak m)$ 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.