Criteria for the presence of the maximal ideal in the set of associated primes

TL;DR

The paper addresses the problem of determining when the maximal ideal $\mathfrak{m}$ lies in the set of associated primes of powers of monomial ideals in $R=K[x_1,\dots,x_n]$. It develops criteria that reduce this question to colon-operations with products of auxiliary variables, yielding equivalences of associated-prime membership between $I^t$ and $I^t:\prod_i y_i^{\alpha_i}$ under explicit hypotheses, and establishing vanishing when the product lies in $I^\ell$ with $\ell\ge t$. These results are distilled into practical corollaries, including a square-free case where $\mathfrak{m}\in\mathrm{Ass}(R/I)$ if and only if $I=\mathfrak{m}$, and a secondary criterion for excluding $\mathfrak{m}$ under alternative generator configurations. The authors bolster their framework with lemmas and propositions and illustrate applications via examples such as wheel graphs and graph-cone constructions, highlighting how depth and embedded primes interact with powers of monomial ideals.

Abstract

In this paper, we establish some criteria to detect the presence of the maximal ideal $(x_1, \ldots, x_n)$ in the set of associated primes of powers of monomial ideals in the polynomial ring $K[x_1, \ldots, x_n]$. Furthermore, for each of these criteria, we illustrate its applicability with corresponding examples.

Theorems & Definitions (23)

• Proposition 2.1
• Definition 2.2
• Theorem 2.5
• Corollary 2.6
• Proposition 2.7
• Theorem 3.1
• proof
• Corollary 3.2
• Proposition 3.3
• proof