Some remarks on the Stanley's depth for multigraded modules

TL;DR

This paper proves Stanley's conjecture for multigraded modules with zero Stanley depth and provides bounds for the Stanley depth of powers of the maximal irrelevant ideal in polynomial rings.

Contribution

It establishes the validity of Stanley's conjecture in a specific case and offers new bounds for Stanley depth of certain ideal powers.

Findings

Stanley's conjecture holds for multigraded modules with sdepth 0

Bounds are provided for the Stanley depth of powers of the maximal irrelevant ideal

The results extend understanding of Stanley depth in multigraded algebra

Abstract

We show that the Stanley's conjecture holds for any multigraded $S$-module $M$ with $\sdepth(M)=0$, where $S=K[x_1,...,x_n]$. Also, we give some bounds for the Stanley depth of the powers of the maximal irrelevant ideal in $S$.