Stabilization of associated prime ideals of monomial ideals -- Bounding the copersistence index

Abstract

The sequence $(\operatorname{Ass}(R/I^n))_{n\in\mathbb{N}}$ of associated primes of powers of a monomial ideal $I$ in a polynomial ring $R$ eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of $I$ after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of $I$. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.