Fluctuations in depth and associated primes of powers of ideals

TL;DR

This paper studies the behavior of associated primes and depth functions of powers of certain monomial ideals, revealing periodicity and providing explicit characterizations that improve on previous bounds.

Contribution

It introduces a new class of monomial ideals, H(r,s), and fully characterizes the depth functions of their powers, showing periodicity and improved variable bounds.

Findings

Depth functions are periodic with period r, repeating m times before stabilizing.

The number of variables needed is lower than in previous general constructions.

Complete characterization of associated primes for powers of the new ideals.

Abstract

We count the numbers of associated primes of powers of ideals as defined by Bandari, Hibi, and Herzog in 2014. We generalize those ideals to monomial ideals $\operatorname{BHH}(m,r,s)$ for $r \ge 2$, $m$, $s \ge 1$; we establish partially the associated primes of powers of these ideals, and we establish completely the depth function of quotients by powers of these ideals: the depth function is periodic of period $r$ repeated $m$ times on the initial interval before settling to a constant value. The number of needed variables for these depth functions are lower than those from general constructions by H\`{a}, Nguyen, Trung, and Trung (2021).