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

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.

💡 Research Summary

The paper investigates the relationship between two important structural properties of standard graded algebras over a field: G‑quadraticity (the existence of a quadratic Gröbner basis after a linear change of coordinates) and the existence of a Koszul filtration (a family of ideals generated by linear forms satisfying certain colon‑ideal conditions). While it is well‑known that a quadratic Gröbner basis guarantees Koszulness, the converse implications are generally strict, and the precise connection between G‑quadraticity and Koszul filtrations has remained unclear.

The authors first recall the necessary background on Gröbner bases, term orders (lexicographic and degree reverse lexicographic), and the definition of Koszul filtrations (KF1–KF3). They then introduce two new combinatorial notions—G‑sets and G‑sequences—tailored to binomial Gröbner bases whose terms have disjoint support. A G‑set is a set of variables such that whenever a variable divides the leading term of a basis element, some (possibly different) variable from the set divides the trailing term. A G‑sequence is a strictly decreasing chain of G‑sets ending in the empty set. These concepts allow the authors to control colon ideals of the form ((G\cup X):x) and to prove Theorem 3.7: if (G) is a reduced quadratic Gröbner basis of binomials with disjoint support and (X) and (X\setminus{x}) are G‑sequences, then ((G\cup X\setminus{x}):x) is again generated by a Gröbner basis of the same type, together with a new G‑sequence (X’). Consequently, when the whole set of variables forms a G‑sequence, one obtains a monomial Koszul filtration of the quotient algebra.

Applying this machinery, the authors prove that for toric ideals equipped with the degree reverse lexicographic order, and for binomial edge ideals, the existence of a quadratic Gröbner basis automatically yields a Koszul filtration (Corollaries 3.8 and 3.10). This resolves a large class of algebras where the two properties coincide.

The paper also contributes practical algorithms for constructing Koszul filtrations. Section 4 presents two procedures: (1) detection of maximal G‑sets and extraction of a G‑sequence from a given Gröbner basis, and (2) iterative construction of the filtration by computing colon ideals ((I:x)) and updating the G‑set accordingly. Implementations in Macaulay2 are provided in the appendix, and the authors demonstrate that the algorithms can handle examples with up to a hundred variables efficiently.

A notable application is to the pinched Veronese algebra. Using the algorithms, the authors show that under the natural coordinates and the degree reverse lexicographic order, this algebra does not admit a quadratic Gröbner basis, confirming that its Koszulness cannot be explained by the standard sufficient condition. This result also illustrates that the existence of a Koszul filtration may fail even for algebras known to be Koszul.

Finally, the authors settle a conjecture of Ene, Herzog, and Hibi (2015) by constructing a G‑quadratic algebra that lacks any Koszul filtration. The example is inspired by recent work on graded Möbius algebras and depends on the characteristic of the base field: in one characteristic the algebra admits a Koszul filtration, while in another it does not. Computational verification via Macaulay2 and the newly developed algorithms confirms the claim. This provides the first explicit counterexample to the conjectured implication “G‑quadratic ⇒ Koszul filtration”.

In summary, the paper clarifies the precise interplay between G‑quadraticity and Koszul filtrations, introduces robust combinatorial tools (G‑sets, G‑sequences) and effective algorithms for constructing filtrations, and applies these results to both classical families (toric, binomial edge) and more exotic examples (pinched Veronese, characteristic‑dependent counterexample). The work advances both the theoretical understanding and the computational practice of Koszul algebras.