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 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 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 R/I and the well- known combinatorial conjecture that every Cohen–Macaulay simplicial complex is partitionable are equivalent.