Equivariant algebraic K-theory of G-rings

TL;DR

This paper develops a functorial, genuine equivariant algebraic K-theory framework for G-rings, unifying various K-theory theories and connecting to significant conjectures and maps in the field.

Contribution

It introduces a new definition of equivariant algebraic K-theory that encodes group actions functorially, enabling a unifying approach to equivariant and classical K-theory.

Findings

Constructs a genuine G-spectrum for algebraic K-theory

Unifies equivariant topological K-theory and Real K-theory

Reinterprets key conjectures and maps in equivariant stable homotopy theory

Abstract

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological $K$-theory, Atiyah's Real $K$-theory, and existing statements about algebraic $K$-theory spectra with $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory.