Strong persistence and associated prime of powers of monomial ideals

TL;DR

This paper investigates the algebraic properties of powers of monomial ideals, proving a key equality for degree 2 ideals, providing a counterexample to a previous conjecture, and addressing the behavior of depth functions.

Contribution

It establishes a new equality for powers of degree ≤ 2 monomial ideals, disproves a related conjecture with a counterexample, and answers a question about depth functions of square-free monomial ideals.

Findings

Proves (I^{k+1}:I)=I^k for degree ≤ 2 monomial ideals

Provides a counterexample to a previous conjecture in the literature

Shows the depth function of some square-free monomial ideals can increase

Abstract

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\leq 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\geq 1$ and we disprove a motivation question that was appeared in \cite[Question 2.51]{CHHV} by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that depth function of square-free monomial ideals are non-increasing.