The behavior of Stanley depth under polarization

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.Comment: Version 2: several proofs were clarified and a minor result was added. Version 3: further improvements based on several readers feedbac