On Coefficient Module of Arbitrary Modules

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 $°(P_{M_k/M}(n))<s-k,$ where $q(M)$ is the relative integral closure of $M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}={\cup }_{n\ge 1}(M{:}_F{\mathfrak{m}}^n)$ 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.