Remarks on the hilbert depth of monomial ideals

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