Triangular objects and systematic K-theory

TL;DR

This paper studies modules over nearly graded rings called systematic rings, unifying and generalizing existing graded and filtered K-theory calculations using idempotent completion and triangular objects in additive categories.

Contribution

It introduces a unified framework for calculating K-theory of systematic rings, extending previous methods through a formalism involving idempotent completion and triangular objects.

Findings

Unified approach to graded and filtered K-theory calculations

Elementary proofs using idempotent completion and triangular objects

Generalization of existing K-theory results for systematic rings

Abstract

We investigate modules over "systematic" rings. Such rings are "almost graded" and have appeared under various names in the literature; they are special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the presence of conditions on the support, and explain how this generalises and unifies calculations of graded and filtered K-theory scattered in the literature. Our treatment makes systematic use of the formalism of idempotent completion and a theory of triangular objects in additive categories, leading to elementary and transparent proofs throughout.