A new proof of the Herzog-Hibi-Zheng theorem

TL;DR

The paper addresses the problem of characterizing when a quadratic monomial ideal $I$ has linear powers, culminating in the HHZ theorem which asserts the equivalence of (a) $I$ having a linear resolution, (b) $I$ having linear powers, and (c) all powers $I^k$ having linear quotients. It presents an elementary proof by leveraging polarization to realize $I^p$ as a cochordal-edge ideal $I(G)$ and then establishes that $I^k$ possesses linear quotients for all $k\ge1$ via a carefully chosen order on standard presentations and a combinatorial even-closed-walk argument. The result provides a short, self-contained alternative to previous proofs and yields corollaries such as that $P^kI^old$ has linear quotients when $I\subset P$ is quadratic with linear resolution, highlighting connections between graph theory and the asymptotic homological behavior of monomial ideals. The work also discusses related questions, including the Nevo–Peeva conjecture, and clarifies implications for Rees algebras and Betti-splitting phenomena.

Abstract

We give a new, elementary proof of the celebrated Herzog-Hibi-Zheng theorem on powers of quadratic monomial ideals.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• proof
• proof : Proof of Theorem \ref{['Thm:HHZ']}
• Corollary 3.1