Alexander duality and Stanley depth of multigraded modules

TL;DR

This paper explores the application of Alexander duality to multigraded modules over polynomial rings, developing tools to analyze Stanley depth and providing partial results supporting Stanley's conjecture.

Contribution

It introduces a novel approach using Alexander duality with different centers and proves Stanley's conjecture for quotients by cogeneric monomial ideals.

Findings

Alexander duality aids in studying Stanley depth.

Stanley's conjecture holds for quotients by cogeneric monomial ideals.

Tools for Stanley's conjecture are further developed.

Abstract

We apply Miller's theory on multigraded modules over a polynomial ring to the study of the Stanley depth of these modules. Several tools for Stanley's conjecture are developed, and a few partial answers are given. For example, we show that taking the Alexander duality twice (but with different "centers") is useful for this subject. Generalizing a result of Apel, we prove that Stanley's conjecture holds for the quotient by a cogeneric monomial ideal.