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Trace methods for equivariant algebraic K-theory

David Chan, Teena Gerhardt, Inbar Klang

TL;DR

The paper builds a comprehensive trace-methods framework for equivariant algebraic $K$-theory by constructing an equivariant Dennis trace to a $G$-equivariant THH variant, $\mathrm{eTHH}$. It develops the norm-oriented interpretation of THH, establishes Morita invariance and additivity for $\mathrm{ETHH}$, and provides a detailed equivariant Dennis trace that connects $K_G$ to $\mathrm{eTHH}$, with fixed-point compatibility aligning with known THH_{C_n} behavior. It also links these constructions to equivariant $A$-theory, showing how coarse and genuine $A$-theory map to free loop spaces on fixed-point spaces, and outlines connections and conjectures about genuine equivariant traces to $\mathrm{ETHH}$. Overall, the work provides foundational tools for computating and comparing equivariant $K$-theory via trace methods, leveraging equivariant norms, spectral Waldhausen categories, and Morita theory. These results open avenues for applying trace techniques to equivariant multiplicative structures and parametrized $h$-cobordism in new settings.

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.

Trace methods for equivariant algebraic K-theory

TL;DR

The paper builds a comprehensive trace-methods framework for equivariant algebraic -theory by constructing an equivariant Dennis trace to a -equivariant THH variant, . It develops the norm-oriented interpretation of THH, establishes Morita invariance and additivity for , and provides a detailed equivariant Dennis trace that connects to , with fixed-point compatibility aligning with known THH_{C_n} behavior. It also links these constructions to equivariant -theory, showing how coarse and genuine -theory map to free loop spaces on fixed-point spaces, and outlines connections and conjectures about genuine equivariant traces to . Overall, the work provides foundational tools for computating and comparing equivariant -theory via trace methods, leveraging equivariant norms, spectral Waldhausen categories, and Morita theory. These results open avenues for applying trace techniques to equivariant multiplicative structures and parametrized -cobordism in new settings.

Abstract

In the past decades, one of the most fruitful approaches to the study of algebraic -theory has been trace methods, which construct and study trace maps from algebraic -theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic -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 -theory. For a finite group, we construct a Dennis trace map from equivariant algebraic -theory to a -equivariant version of topological Hochschild homology; for 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 -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.
Paper Structure (20 sections, 58 theorems, 189 equations)

This paper contains 20 sections, 58 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Background
  5. Equivariant homotopy theory and norms
  6. Fixed points
  7. The norm
  8. Homotopy theory of orthogonal $G$-spectra
  9. Equivariant algebraic $K$-theory
  10. Twisted topological Hochschild homology
  11. Spectral categories
  12. Equivariant topological Hochschild homology
  13. Hochschild homology theories for $G$-spectral categories
  14. Fixed points of the cyclic nerve
  15. A model structure on $G$-spectral categories
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $R$ be a cofibrant $C_{n}$-ring spectrum, and let $\Delta_{C_n} \leq C_n \times S^1$ denote the diagonal subgroup isomorphic to $C_n$. There is an isomorphism of orthogonal $C_n$-spectra

Theorems & Definitions (175)

  • Theorem 1.1: \ref{['thm: geometric fixed points of ETHH is twisted THH for rings']}
  • Theorem 1.2: \ref{['theorem: Morita invariance']} and \ref{['cor-ETHH-additivity']}
  • Theorem 1.3: \ref{['cor-trace-R']}
  • Theorem 1.4: \ref{['thm-trace-to-twisted']}
  • Proposition 1.1: \ref{['prop-trace-A-loop']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1: mandell-may
  • Remark 2.1
  • Definition 2.3
  • ...and 165 more