Abstract
Let K be a field, R=K[X1,…,Xn] be the polynomial ring and J⊊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 Ip 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.
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