Constructing Koszul filtrations: existence and non-existence for G-quadratic algebras

TL;DR

This work probes when $G$-quadratic algebras admit Koszul filtrations and how this relates to Koszulness. It introduces $G$-sets and $G$-sequences to build monomial Koszul filtrations from quadratic binomial Gröbner bases and develops Macaulay2–based algorithms to construct or refute filtrations. The authors prove that a quadratic Gröbner basis induces a Koszul filtration for toric algebras under degree reverse lexicographic order and for binomial edge ideals, and they resolve Ene–Hibi's conjecture by constructing a G-quadratic algebra with no Koszul filtration, including characteristic-dependent phenomena. They apply the methods to the pinched Veronese algebra to give a new argument that it has no quadratic Gröbner basis in natural coordinates and no monomial Koszul filtration, and they present a concrete counterexample—the Möbius algebra of a broken 4-trampoline graph—that is G-quadratic yet lacks a Koszul filtration over $ ext{F}_3$, highlighting limitations of the filtration paradigm.

Abstract

Given a standard graded algebra over a field, we consider the relationship between G-quadraticity and the existence of a Koszul filtration. We show that having a quadratic Gröbner basis implies the existence of a Koszul filtration for toric algebras equipped with the degree reverse lexicographic term order and for algebras defined by binomial edge ideals. We also resolve a conjecture of Ene, Herzog, and Hibi by constructing an example where this implication fails. These results are underpinned by algorithms we develop for constructing Koszul filtrations. We also demonstrate the utility of these algorithms on the pinched Veronese algebra.

Figures (1)

• Figure 1: A labelling of the broken 4-trampoline corresponding to the presentation of the algebra $B$.

Theorems & Definitions (46)

• Theorem 1.2
• Theorem 1.3
• Definition 3.1
• Definition 3.2
• Example 3.3
• Example 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof