On Quillen's calculation of graded $K$-theory

TL;DR

This paper extends Quillen's method for calculating graded K-groups of Z-graded rings to more general gradings involving arbitrary groups, enabling calculations for Z^m-graded rings.

Contribution

It generalizes Quillen's calculation to include gradings in a product of Z and any group G, facilitating the computation of graded K-theory for Z^m-graded rings.

Findings

Extended Quillen's calculation to Z imes G gradings

Enabled inductive calculations for Z^m-graded rings

Provided a framework for computing graded K-theory in more complex settings

Abstract

We adapt Quillen's calculation of graded K-groups of Z-graded rings with support in N to graded K-theory, allowing gradings in a product Z \times G with G an arbitrary group. This in turn allows us to use inductions and calculate graded K-theory of Z^m-graded rings. Here Z is the ring of integers and N positive natural numbers.