On the K-theory of $\mathbb{Z}$-categories

TL;DR

This paper explores the K-theory of Z-linear categories, establishing key connections with ring theory and proving significant results like vanishing, fundamental theorems, and homotopy invariance.

Contribution

It introduces new links between categorical properties and ring-theoretic notions, leading to important K-theory results for Z-linear categories.

Findings

Negative K-theory vanishing result

Fundamental theorem for Z-linear categories

Homotopy invariance of K-theory

Abstract

We establish connections between the concepts of Noetherian, regular coherent, and regular n-coherent categories for Z-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative K-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the K-theory of Z-linear categories.