$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.

💡 Research Summary

The paper investigates the homological relationship between a finite‑dimensional algebra (A=KQ/I) and its associated monomial algebra (A_{\text{mon}}=KQ/\langle\operatorname{tip}(I)\rangle), focusing on the case where at least one of them is (d)-Koszul. After recalling the definition of a (d)-Koszul algebra—an algebra whose minimal graded projective resolution of the simple module (K) has generators only in degrees (di) or (di+1)—the authors consider Gröbner bases for the ideal (I). They prove that if (I) admits a reduced Gröbner basis consisting solely of homogeneous elements of degree (d) (i.e., (I) is (d)-homogeneous), then (A) is (d)-Koszul if and only if its monomial counterpart (A_{\text{mon}}) is (d)-Koszul. The proof hinges on the observation that the leading‑term ideal (\langle\operatorname{tip}(I)\rangle) captures all critical overlaps of the relations, so the homological shape of the minimal resolution is preserved under passage to the monomial algebra.

Motivated by this equivalence, the authors introduce two new classes of algebras. An algebra is called 2‑(d)‑determined when its defining ideal is generated by elements of degree (d) and (d+1) only, and the minimal projective resolution of (K) has generators confined to degrees (di) or (di+1). A 2‑(d)‑Koszul algebra is a 2‑(d)‑determined algebra that additionally satisfies the stronger Ext‑condition (\operatorname{Ext}^i_A(K,K)) being non‑zero only in degrees (di) or (di+1). The paper shows that every 2‑(d)‑determined monomial algebra is automatically 2‑(d)‑Koszul. This is achieved by a detailed analysis of critical pairs in the monomial setting: when all overlaps of the monomial relations resolve trivially, the resolution of the simple module exhibits the required degree pattern, and the Ext‑algebra inherits the same regularity.

A central structural result describes the ideal of relations for a 2‑(d)‑determined monomial algebra. Such an ideal can be written as \