Trace methods for equivariant algebraic K-theory

TL;DR

This paper develops trace methods for equivariant algebraic K-theory, constructing trace maps to equivariant topological Hochschild homology and establishing foundational properties for computational and theoretical advances.

Contribution

It introduces a Dennis trace map for equivariant algebraic K-theory, extending classical trace methods to the equivariant setting and analyzing its properties.

Findings

Constructed a Dennis trace map for finite group actions.

Revealed fixed point properties linking to known trace maps.

Established Morita invariance and multiplicative norm properties of equivariant THH.

Abstract

In the past decades, one of the most fruitful approaches to the study of algebraic $K$-theory has been trace methods, which construct and study trace maps from algebraic $K$-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic $K$-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic $K$-theory. For $G$ a finite group, we construct a Dennis trace map from equivariant algebraic $K$-theory to a $G$-equivariant version of topological Hochschild homology; for $G$ the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant $A$-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm.