Equivariant localizing motives and multiplicative norms on algebraic K-theory

TL;DR

This paper develops a theory of equivariant algebraic K-theory with multiplicative norms, extending noncommutative motives to the equivariant setting and constructing a genuine equivariant THH with a compatible Dennis trace map.

Contribution

It introduces a genuine equivariant framework for algebraic K-theory and THH, generalizing noncommutative motives to incorporate equivariance and multiplicative structures.

Findings

Constructed multiplicative norms on equivariant algebraic K-theory.

Established a genuine equivariant version of THH with Dennis trace map.

Proved that norms of stable categories preserve equivariant motivic equivalences.

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.