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

TL;DR

This paper explores the conditions under which G-quadratic algebras possess Koszul filtrations, providing new algorithms for their construction and presenting counterexamples that clarify the relationship between these properties.

Contribution

It establishes when quadratic Gr"obner bases imply Koszul filtrations in certain algebras and resolves a conjecture by constructing a counterexample.

Findings

Quadratic Gr"obner basis implies Koszul filtration for specific algebras

Counterexample showing the implication does not always hold

Algorithms developed for constructing Koszul filtrations

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 Gr\"obner 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.