The Stanley Depth in the Upper Half of the Koszul Complex

TL;DR

This paper proves that the $k$-th syzygy module of the residue field in a polynomial ring has Stanley depth $n-1$ for a specific range of $k$, confirming a conjecture and expanding known cases.

Contribution

It confirms a conjecture by Bruns et al. that the Stanley depth of certain syzygy modules is $n-1$, providing new insights into modules with higher-dimensional graded components.

Findings

Stanley depth of $k$-th syzygy module is $n-1$ for $loor{n/2} \, extless k extless n$

Provides the Stanley depth for modules with graded components of dimension greater than 1

Uses analysis of a matching in the Boolean algebra to prove the result

Abstract

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than $1$. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.