Stanley depth and size of a monomial ideal

TL;DR

This paper explores the relationship between the size of monomial ideals and their Stanley depth, establishing new bounds and applying duality to derive regularity estimates.

Contribution

It demonstrates that size provides a lower bound for Stanley depth and uses Alexander duality to find upper bounds for regularity of squarefree monomial ideals.

Findings

Size is a lower bound for Stanley depth.

Upper bounds for regularity are obtained via Alexander duality.

The results extend understanding of monomial ideal invariants.

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. We show that the size is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.