Remarks on the hilbert depth of monomial ideals

Abstract

Let K be a infinite field, S = K[x ₁, …, x_n ] and 0 ⊂ I ⊊ J ⊂ S two monomial ideals. In this note, we prove that if S/I is Cohen Macaulay, then hdepth (I) ≥ hdepth (S/I) + 1. Moreover, we reobtain the following basic properties of the Hilbert depth, namely depth (J/I) ≤ hdepth (J/I) ≤ dim (J/I). The main tools used are polarization and the Stanley-Reisner correspondence between (relative) simplicial complexes and (quotients of) squarefree monomial ideals.