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

2026-02-06

AI Summary

This study addresses the question of whether G-quadratic algebras always admit a Koszul filtration. By integrating Gröbner basis theory, combinatorial commutative algebra, and a newly devised algorithm for constructing Koszul filtrations, the authors establish—for the first time—that in toric algebras and binomial edge ideal algebras, G-quadraticity with respect to the degree reverse lexicographic order indeed implies the existence of a Koszul filtration. Moreover, they construct explicit counterexamples that disprove the conjecture of Ene–Herzog–Hibi asserting the general validity of this implication. The successful application of the proposed algorithm to pinched Veronese algebras further demonstrates its efficacy and extends the current theoretical boundaries in the field.

Technology Category

Knowledge Representation and Reasoning: Qualitative ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial Optimization

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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.

Problem

• Koszul filtration
• G-quadratic algebra
• quadratic Gröbner basis
• toric algebra
• binomial edge ideal

Innovation

• Koszul filtration
• G-quadratic algebra
• quadratic Gröbner basis
• toric algebra
• binomial edge ideal