Comparing Hilbert depth of $I$ with Hilbert depth of $S/I$. III

TL;DR

This paper investigates the relationship between the Hilbert depth of a squarefree monomial ideal and its quotient, establishing bounds under specific conditions to deepen understanding of their algebraic properties.

Contribution

It proves that for certain cases, the Hilbert depth of the ideal is at least one less than that of the quotient, extending previous knowledge on depth relations.

Findings

If hdepth(S/I) ≤ 8, then hdepth(I) ≥ hdepth(S/I) - 1.

If the number of variables n ≤ 10, the same bound holds.

Provides new bounds connecting Hilbert depths of ideals and their quotients.

Abstract

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 8$ or $n\leq 10$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)-1$.