Koszul-Like Algebras and Modules

In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin–Schelter regular algebras of global dimension $5$ as special examples. Basic properties of Koszul-like modules are discussed. In particular, some necessary and sufficient conditions for $\mathcal{KL}(A)=\mathcal{L}(A)$ are provided, where $\mathcal{KL}(A)$ and $\mathcal{L}(A)$ denote the categories of Koszul-like modules and modules with linear presentations (see [1]–[3], etc.) respectively, and $A$ is a Koszul-like algebra. We construct new Koszul-like algebras from the known ones by the “one-point extension”. Some criteria for a graded algebra to be Koszul-like are provided. Finally, we construct many classical Koszul objects from the given Koszul-like objects.