# Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals

## TL;DR

This paper establishes an upper bound on the stability index for the depth of integral closures of powers of monomial ideals and classifies when these closures become Cohen-Macaulay for large powers.

## Contribution

It provides a new upper bound on the depth stability index of integral closures of monomial ideals and characterizes ideals with eventually Cohen-Macaulay integral closures.

## Key findings

- Upper bound on $ar{ ext{dstab}}(I)$ in terms of ring dimension and generating degree
- Classification of monomial ideals with Cohen-Macaulay integral closures for large powers
- Depth of $R/ar{I^n}$ stabilizes beyond a certain power

## Abstract

Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.