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 subset of I be two monomial ideals in R. In this paper we show that sdepth I/J - depth I/J = sdepth I-p/J(p) - depth I-p/J(p), where sdepth I/J denotes the Stanley depth and I-p denotes Polarization the polarization. This solves a conjecture by Herzog [9] 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 Rh I and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent. (C) 2015 Elsevier Inc. All rights reserved.