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

TL;DR

The paper investigates Hilbert depth as an invariant of monomial ideals, connecting it to depth and Stanley depth via a combinatorial framework that reduces to squarefree cases through polarization. It proves a central equivalence: $I$ is principal iff $\operatorname{hdepth}(I)=n$ and $\operatorname{hdepth}(S/I)=n-1$, and develops a Kruskal–Katona–based criterion to compare $\operatorname{hdepth}(I)$ with $\operatorname{hdepth}(S/I)$. The authors establish sharp bounds showing $\operatorname{hdepth}(I)\ge\operatorname{hdepth}(S/I)+1$ when $I$ is squarefree and $\operatorname{hdepth}(S/I)\le 3$ (and for $n\le 5$), while also proving several results where $\operatorname{hdepth}(I)\ge\operatorname{hdepth}(S/I)$ in other regimes, complemented by explicit counterexamples that delineate the method’s limits. A detailed analysis of the case $\operatorname{hdepth}(S/I)=5$ further solidifies the relationship in that regime and yields conjectures for broader validity, with practical implications for understanding the depth landscape of monomial ideals.

Abstract

Let $I$ be a monomial ideal of $S=K[x_1,\ldots,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\operatorname{hdepth}(I)=n$, (iii) $\operatorname{hdepth}(S/I)=n-1$. Assuming that $I$ is squarefree, we prove that if $\operatorname{hdepth}(S/I)\leq 3$ or $n\leq 5$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$. Also, we prove that if $\operatorname{hdepth}(S/I)\leq 5$ or $n\leq 7$ then then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)$.

Theorems & Definitions (54)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4