# $k$-clean monomial ideals

## TL;DR

This paper introduces $k$-clean monomial ideals, extending the concept of clean ideals, and explores their properties, showing their relation to $k$-decomposability of simplicial complexes and identifying classes of ideals that are $k$-clean.

## Contribution

It defines $k$-clean monomial ideals, establishes their properties, and connects them to $k$-decomposability and various classes of monomial ideals, broadening the understanding of their structure.

## Key findings

- $k$-clean ideals generalize clean ideals with hierarchical properties.
- A $(d-1)$-dimensional simplicial complex is $k$-decomposable iff its Stanley-Reisner ideal is $k$-clean.
- Classes like monomial complete intersections and Cohen-Macaulay ideals of codimension 2 are $k$-clean for all $k",

## Abstract

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show that a $(d-1)$-dimensional simplicial complex is $k$-decomposable if and only if its Stanley-Reisner ideal is $k$-clean, where $k\leq d-1$. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are $k$-clean for all $k\geq 0$.