The behavior of Stanley depth under polarization

Abstract

Let $K$ be a field, $R=K[X_1,...,X_n]$ be the polynomial ring and $J⊊I$ 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.

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

References in corpus (3)

• Stanley decompositions and localization
• An algorithm for computing the multigraded Hilbert depth of a module
• An algorithm to compute the Hilbert depth

Cited by in corpus (2)

• Stanley depth and simplicial spanning trees
• Stanley depth of monomial ideals