On Coefficient Module of Arbitrary Modules

TL;DR

This paper investigates the structure of coefficient modules associated with arbitrary modules over Noetherian local rings, establishing existence, uniqueness, and properties of these modules, with applications to module theory.

Contribution

It introduces a new framework for coefficient modules of arbitrary modules, including existence, uniqueness, and structural theorems, extending previous work to more general modules.

Findings

Existence of a unique largest module with finite length quotient

Structural theorem for these modules

Existence of coefficient modules between specific bounds

Abstract

Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by Herzog-Puthenpurakal-Verma in \cite{herzog}, we show the existence of an unique largest $R$-module $M_{k}$ with $\ell_R(M_{k}/M)<\infty$ and $M\subseteq M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that $\deg(P_{M_{k}/M}(n))<s-k,$ where $q(M)$ is the relative integral closure of $M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}=\cup_{n\geq 1}(M:_F\mathfrak{m}^n)$ is the saturation of $M$. We also provide a structure theorem for these modules. Furthermore, we establish the existence of coefficient modules between $I(M)M$ and $M$, where $I(M)$ denotes the $0$-th Fitting ideal of $F/M$, and discuss their structural properties. Finally, we present some applications and discuss some properties.