Stanley depth and Stanley support-regularity of monomial ideals

Abstract

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by \(1\) is a lower bound for its depth. Associated with this, Herzog, Popescu and Vladoiu showed that the size increased by \(1\) is simultaneously a lower bound for its Stanley depth. It turns out that the splitting-variables technique in their paper is particularly suited for comparing the Stanley depth and size of squarefree monomial ideals. In this paper, we generalize this technique and investigate the quotient \(I/J\) when \(J\subsetneq I\) are general monomial ideals. As an application, we consider lower bounds for Stanley depth of quotients of monomial ideals under various conditions. Through Alexander duality, we also consider upper bounds for the Stanley support-regularity of edge ideals of clutters. Meanwhile, recently, Katthan and Seyed Fakhari established another lower bound for depth and Stanley depth of \(I/J\) in terms of the lcm number. Using the splitting-variables technique, we provide an alternate proof of their result.