Constructing indecomposable integrally closed modules over a two-dimensional regular local ring

TL;DR

This paper constructs a broad class of indecomposable integrally closed modules of rank two over two-dimensional regular local rings, explicitly from monomial ideals, and investigates their indecomposability, addressing a question by Kodiyalam.

Contribution

It introduces a method to explicitly construct indecomposable integrally closed modules from monomial ideals, expanding understanding of their structure and properties.

Findings

Constructed a large class of indecomposable integrally closed modules.

Showed these modules have non-simple Fitting ideals.

Provided an answer to Kodiyalam's question.

Abstract

In this article, we construct integrally closed modules of rank two over a two-dimensional regular local ring. The modules are explicitly constructed from a given complete monomial ideal with respect to a regular system of parameters. Then we investigate their indecomposability. As a consequence, we have a large class of indecomposable integrally closed modules whose Fitting ideal is not simple. This gives an answer to Kodiyalam's question.