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On coefficient modules of arbitrary modules
Part of:
Local rings and semilocal rings
Published online by Cambridge University Press: 29 August 2025
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 [10], we show the existence of a unique largest R-module Mk with 
$\ell_R(M_{k}/M) \lt \infty$ and 
$M\subseteq M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that 
$\deg(P_{M_{k}/M}(n)) \lt s-k,$ where q(M) is the relative integral closure of 
$M,$ defined by 
$q(M):=\overline{M}\cap M^{sat},$ where 
$M^{sat}=\cup_{n\geqslant 1}(M:_F\mathfrak{m}^n)$ 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 0th Fitting ideal of 
$F/M$, and discuss their structural properties. Finally, we present some applications and discuss some properties.
MSC classification
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.
Footnotes
*
This work was partially supported by CNPq-Brazil-Grants 309316/2011-1, and FAPESP Grant 2012/20304-1.
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