The behavior of Stanley depth under polarization

TL;DR

This paper proves a key relation between Stanley depth and polarization of monomial ideals, confirming a conjecture and linking Stanley's conjecture to the squarefree case, with implications for Cohen-Macaulay complexes.

Contribution

It establishes the equality of Stanley depth and depth differences under polarization, confirming a conjecture and reducing Stanley's conjecture to the squarefree case.

Findings

Proves the equality of sdepth and depth differences under polarization.

Reduces Stanley's conjecture to the squarefree case.

Links Stanley's conjecture to Cohen-Macaulay simplicial complexes.

Abstract

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\ {I^p/J^p}$, where $\mathrm{sdepth}\ I/J$ denotes the Stanley depth and $I^p$ denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form $I/J$) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form $R/I$ and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.