The behavior of Stanley depth under polarization

Let KK be a field, R=K[X1,…,Xn]R=K[X1,…,Xn] be the polynomial ring and J⊊IJ⊊I be two monomial ideals in R. In this paper we show thatsdepthI/J−depthI/J=sdepthIp/Jp−depthIp/Jp, where sdepthI/J denotes the Stanley depth and IpIp denotes the polarization. This solves a conjecture by Herzog [9] and reduces the famous Stanley conjecture (for modules of the form I/JI/J) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R/IR/I and the well-known combinatorial conjecture that every Cohen–Macaulay simplicial complex is partitionable are equivalent.