On strongly Koszul algebras and tidy Gröbner bases

Strongly Koszul algebras were introduced by Herzog, Hibi and Restuccia in 2000. The goal of the present paper is to provide an in-depth study of such algebras and to investigate how strong Koszulness interacts with the existence of a quadratic Gröbner basis for the defining ideal. Firstly, we prove that the existence of a quadratic revlex-universal Gröbner basis with a strong sparsity condition (that we name “tidiness”) is a sufficient condition for strong Koszulness, and exhibit several concrete examples arising from determinantal objects and Macaulay’s inverse system. We then prove that there exist standard graded algebras that are strongly Koszul but do not admit a Gröbner basis of quadrics even after a linear change of coordinates, thus answering negatively a question posed by Conca, De Negri and Rossi. As a bonus, we prove that strong Koszulness behaves well under tensor and fiber products of algebras and illustrate how Severi varieties and Macaulay’s inverse system interact to produce examples of strongly Koszul algebras with a geometric flavor.

💡 Research Summary

The paper undertakes a systematic study of strongly Koszul algebras, a refinement of the classical Koszul property introduced by Herzog, Hibi, and Restuccia in 2000. Using the definition of Conca–De Negri–Rossi, a standard graded k‑algebra R is called strongly Koszul if there exists a k‑basis B of R₁ such that for every proper subset A⊂B and every element b∈B∖A, the colon ideal (a | a∈A):R b is generated by elements of B. This definition coincides with the original one in the toric case, but it is not known whether the two notions diverge in general.

The first major result (Theorem A, see Theorem 2.7) shows that strong Koszulness is preserved under both tensor products and fiber products of standard graded algebras. The authors introduce natural injections ι₁, ι₂ of the factors into the product and prove, via a series of lemmas on colon ideals and Tor‑vanishing, that the product is strongly Koszul with respect to the union of the lifted bases if and only if each factor is strongly Koszul with respect to its own basis. This extends earlier work of Herzog–Hibi–Restuccia beyond the toric setting.

The core technical innovation is the notion of a “tidy polynomial.” A homogeneous polynomial f∈k