Results on the normality of square-free monomial ideals and cover ideals under some graph operations

TL;DR

This paper develops methods to generate normal square-free monomial ideals from existing ones and examines how certain graph operations affect the normality of cover ideals, revealing that linear combinations of normal ideals may not remain normal.

Contribution

It introduces new techniques for constructing normal ideals and analyzes the impact of graph operations on the normality of cover ideals, including cases where normality is not preserved.

Findings

Linear combinations of two normal ideals may not be normal.

The paper investigates the integral closedness of all powers of certain non-normal ideals.

Techniques for producing normal ideals from existing ones are established.

Abstract

In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.