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Equivariant localizing motives and multiplicative norms on algebraic K-theory

Kaif Hilman, Maxime Ramzi

Abstract

We construct multiplicative norms on equivariant nonconnective algebraic $K$-theory for finite groups $G$. We also construct a genuine equivariant version of THH equipped with a Dennis trace map from K-theory compatible with the multiplicative norms. To do so, we follow the general strategy of Blumberg-Gepner-Tabuada in the nonequivariant case by generalizing their category of localizing motives to the genuine equivariant context, building upon the theory of perfect $G$-stable categories of the first-named author. Crucially, we proceed using the recent perspective on noncommutative motives by the second-named author with Sosnilo and Winges which allows us to deal with non-exact functors on this category of motives. Together with an isotropy separation argument for equivariant cubes, we prove our main theorem that norms of stable categories preserve equivariant motivic equivalences. As an immediate consequence, we obtain a unique equivariant multiplicative refinement of nonconnective algebraic $K$-theory. From these constructions and results, we draw several applications, namely: (1) that the endofunctor of (equivariant) tensor powers on ordinary perfect stable categories preserve motivic equivalences; (2) that the multiplicative norms also preserve the additive motivic equivalences, thus yielding a motivic refinement of a result of Elmanto-Haugseng and Cnossen-Haugseng-Lenz-Linskens that connective algebraic K-theory admits multiplicative norms; (3) we construct a genuine equivariant version of topological Hochschild homology equipped with a Dennis trace map that is compatible with multiplicative norms; and (4) we prove that every genuine $G$-spectrum is the K-theory of a perfect $G$-stable category.

Equivariant localizing motives and multiplicative norms on algebraic K-theory

Abstract

We construct multiplicative norms on equivariant nonconnective algebraic -theory for finite groups . We also construct a genuine equivariant version of THH equipped with a Dennis trace map from K-theory compatible with the multiplicative norms. To do so, we follow the general strategy of Blumberg-Gepner-Tabuada in the nonequivariant case by generalizing their category of localizing motives to the genuine equivariant context, building upon the theory of perfect -stable categories of the first-named author. Crucially, we proceed using the recent perspective on noncommutative motives by the second-named author with Sosnilo and Winges which allows us to deal with non-exact functors on this category of motives. Together with an isotropy separation argument for equivariant cubes, we prove our main theorem that norms of stable categories preserve equivariant motivic equivalences. As an immediate consequence, we obtain a unique equivariant multiplicative refinement of nonconnective algebraic -theory. From these constructions and results, we draw several applications, namely: (1) that the endofunctor of (equivariant) tensor powers on ordinary perfect stable categories preserve motivic equivalences; (2) that the multiplicative norms also preserve the additive motivic equivalences, thus yielding a motivic refinement of a result of Elmanto-Haugseng and Cnossen-Haugseng-Lenz-Linskens that connective algebraic K-theory admits multiplicative norms; (3) we construct a genuine equivariant version of topological Hochschild homology equipped with a Dennis trace map that is compatible with multiplicative norms; and (4) we prove that every genuine -spectrum is the K-theory of a perfect -stable category.
Paper Structure (23 sections, 52 theorems, 65 equations)

This paper contains 23 sections, 52 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Applications
  4. Relation to other work
  5. Organization of the paper
  6. Acknowledgements
  7. Elements
  8. Preliminaries
  9. Perfect --stable categories and Karoubi sequences
  10. Isotropy separation for fully faithfulness
  11. Equivariant localizing motives
  12. Definitions and constructions
  13. --symmetric monoidality of localizing motives
  14. Corepresentedness of localizing equivariant K--theory
  15. Applications
  16. ...and 8 more sections

Key Result

Theorem 1

Let $H\leq K$ be an inclusion of subgroups of a finite group $G$. The multiplicative norm functor $\mathrm{N}_H^K : \mathrm{Cat}^{\mathrm{perf}}_H\to\mathrm{Cat}^{\mathrm{perf}}_K$ sends $H$-motivic equivalences to $K$-motivic equivalences.

Theorems & Definitions (135)

  • Theorem 1: cf. \ref{['cor:Nmweq']}
  • Theorem 2: cf. \ref{['thm:GMotGPres', 'thm:motnorm', 'prop:corep', 'cor:consKlaxG']}
  • Corollary 3: cf. \ref{['cor:tensor_power_preserve_moteq']}
  • Theorem 4: cf. \ref{['cor:norms_on_splitting_motives']}
  • Theorem 5: cf. \ref{['cor:dennis_trace']}
  • Theorem 6: cf. \ref{['thm:every_spectrum_is_K-theory']}
  • Example 2.1.7
  • Definition 2.1.10
  • Example 2.1.11
  • Proposition 2.1.12
  • ...and 125 more