Indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring
type: Item
instance of: scholarly article
title: Indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring
zbMATH Open document ID: 1490.13012
author: Futoshi Hayasaka
published in: Journal of Pure and Applied Algebra
publication date: 2022-05-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/2112.02885
review text: In this review the following notations are used: \((R, \mathfrak m)\) is a two-dimensional regular local ring with infinite residue field, \(K\) its field of quotients. Let \(\mathfrak m:=(x,y)\); \(\mathcal I\) is the set of monomial ideals -- with respect to \(\{x,y\}\); \(\mathcal M\) is the family of finitely generated torsion-free \(R\)-modules. For \(M \in \mathcal M\) let \(\overline M\subset M\otimes_RK\) be the integral closure of \(M\) in the sense of \textit{D. Rees} [Math. Proc. Camb. Philos. Soc. 101, 431--449 (1987; Zbl 0628.13004)] and Chapter 6 in [\textit{I. Swanson} and \textit{C. Huneke}, Integral closure of ideals, rings, and modules. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)]. The author continues his work on modules in \(\mathcal M\) [\textit{F. Hayasaka} and \textit{E. Hyry}, Proc. Am. Math. Soc. 138, No. 2, 545--551 (2010; Zbl 1185.13028); \textit{F. Hayasaka}, J. Algebra 556, 879--907 (2020; Zbl 1440.13038)]. Some of the results proved in these papers are needed in order to prove the main result which is stated below. For \(M\in\mathcal M\) let \(I(M)\) be the ideal \(I(M):=\mathrm{Fitt}_0(M^{**}/M)\); note that \(I(\overline M)=\overline{I(M)}\). \textit{V. Kodiyalam} [Trans. Am. Math. Soc. 347, No. 9, 3551--3573 (1995; Zbl 0848.13007)] and \textit{V. Kodiyalam} and \textit{R. Mohan} [J. Algebra 425, 392--409 (2015; Zbl 1318.13012)] studied modules \(M\) in \(\mathcal M\) and their integral closure \(\overline M\). In particular: Assume that \(M\) has no free summand. In this case Kodiyalam -- in the first paper mentioned above -- showed: \textit{If the ideal \(\overline{I(M)}\) is simple, then \(\overline M\) is indecomposable.} He asked if the converse to this theorem holds also, i.e., let \(M=\overline M\) be an indecomposable \(R\)-module of rank larger than 1. Does this imply that \(I(M)\) is a simple complete ideal? A particular monomial ideal plays an important role. Let \(r\geq 2\) be an integer. Let \( a :=(a_0,\…
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 7517052
zbMATH Keywords: integral closure
zbMATH Keywords: indecomposable module
zbMATH Keywords: monomial ideal
zbMATH Keywords: regular local ring
reviewed by: Karl-Heinz Kiyek
MaRDI profile type: Publication
OpenAlex ID: W4200634184
arXiv ID: 2112.02885
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DOI: 10.1016/J.JPAA.2022.107026
description: scientific article