K-theory of n-coherent rings

TL;DR

This paper investigates the algebraic K-theory of strong n-coherent rings, establishing equivalences of K-groups and analyzing nilpotent K-theory components under certain finiteness conditions.

Contribution

It proves that for strong n-coherent rings with finitely n-presented modules of finite projective dimension, the K-theory of the ring equals that of a specific exact subcategory.

Findings

$ ext{FP}_n(R)$ is an exact category.

$K_i(R) = K_i( ext{FP}_n(R))$ for all } i \

Expression of $ ext{Nil}_i(R)$ in terms of the category.

Abstract

Let $R$ be a strong $n$-coherent ring such that each finitely $n$-presented $R$-module has finite projective dimension. We consider $\mathcal{FP}_{n}(R)$ the full subcategory of $R$-Mod of finitely $n$-presented modules. We prove that $\mathcal{FP}_{n}(R)$ is an exact category, $K_{i}(R) = K_{i}(\mathcal{FP}_{n}(R))$ for every $i\geq 0$ and obtain an expression of $\operatorname{Nil}_{i}(R)$.