On $(n,d)$-Coherence and Its Applications to Algebraic $\mathsf{K}$-Theory

Abstract

We introduce a unified approach to finiteness conditions in homological algebra and algebraic $\mathsf{K}$-theory by studying the class of $(n,d)$-coherent rings, defined for $n\in {\mathbb{N}}^{\ast }$ and $d\in {\mathbb{N}}^{\ast }\cup \{\infty \}$. This framework simultaneously controls higher finiteness conditions and bounds on the projective dimensions of modules. In the context of algebraic $\mathsf{K}$-theory, we prove that, for a left $(n,d)$-coherent ring, the inclusion $\mathsf{proj}(R)\hookrightarrow {\mathsf{FP}}_n^{\le d}(R)$ induces isomorphisms on all nonnegative $\mathsf{K}$-groups. We then introduce the corresponding relative notions of ${\mathsf{FP}}_n^{\le d}$-injective, ${\mathsf{FP}}_n^{\le d}$-projective, ${\mathsf{FP}}_n^{\le d}$-flat, and ${\mathsf{FP}}_n^{\le d}$-cotorsion modules, and obtain characterizations of $(n,d)$-coherent rings in terms of these classes. When $d\geq\gD(R)$ or $d=\infty$, our notions recover several previously studied classes and results.