$(n,d)$-Coherent Rings

We investigate finiteness conditions on modules of bounded projective dimension and their connection with generalized notions of coherence. For a ring $R$, we consider the class ${\mathsf{F}\mathsf{P}}_n^{\le d}(R)$ of finitely $n$-presented modules of projective dimension at most $d$ and develop the corresponding relative homological theory. We establish several characterizations of left $(n,d)$-coherent rings in the sense of Mao and Ding [43], in terms of ${\mathsf{F}\mathsf{P}}_n^{\le d}(R)$ and the associated classes of ${\mathsf{F}\mathsf{P}}_n^{\le d}$-injective, ${\mathsf{F}\mathsf{P}}_n^{\le d}$-projective, ${\mathsf{F}\mathsf{P}}_n^{\le d}$-flat, and ${\mathsf{F}\mathsf{P}}_n^{\le d}$-cotorsion modules. As a consequence, when $d\ge \text{gD}(R)$ or $d=\mathrm{\infty }$, we recover Costa’s $n$-coherence [17] and obtain new characterizations of regularly coherent rings.