Rigidity of the category of localizing motives

TL;DR

This work establishes that the category ${\operatorname{Mot}}^{\operatorname{loc}}$ of localizing motives is rigid (hence dualizable) and extends rigidity to motivic categories ${\operatorname{Mot}}^{\operatorname{loc}}_{\mathcal{E}}$ over rigid bases. It develops a robust internal-Hom and morphism calculus in these motivic categories, proving corepresentability results for topological cyclic and restriction homologies in the connective ${\mathbb{E}}_{1}$-setting, and deriving a module structure on nil K-theory via TR. The authors introduce refined cyclic theories, analyze invariance under ${\mathbb{A}}^{1}$-localization, and provide detailed applications to K-homology of schemes and equivariant motives, including Farrell–Jones type reformulations. The results yield deep structural insights into noncommutative motives, enabling explicit computations (e.g., TR, TC) and illuminating the interaction between algebraic K-theory, cyclic homology, and motivic localization in a highly structured, higher-categorical framework.

Abstract

In this paper we study the category of localizing motives $\operatorname{Mot}^{\operatorname{loc}}$ -- the target of the universal finitary localizing invariant of idempotent-complete stable categories as defined by Blumberg-Gepner-Tabuada. We prove that this (presentable stable) category is rigid symmetric monoidal in the sense of Gaitsgory and Rozenblyum. In particular, it is dualizable. More precisely, we prove a more general version of this result for the category $\operatorname{Mot}^{\operatorname{loc}}_{\mathcal{E}}$ -- the target of the universal finitary localizing invariant of dualizable modules over a rigid symmetric monoidal category $\mathcal{E}.$ We obtain general results on morphisms and internal $\operatorname{Hom}$ in the categories $\operatorname{Mot}^{\operatorname{loc}}_{\mathcal{E}}$ of localizing motives. As an application we compute the morphisms in multiple non-trivial examples. In particular, we prove the corepresentability statements for $\operatorname{TR}$ (topological restriction) and $\operatorname{TC}$ (topological cyclic homology) when restricted to connective $\mathbb{E}_1$-rings. As a corollary, for a connective $\mathbb{E}_{\infty}$-ring $R$ we obtain a $\operatorname{TR}(R)$-module structure on the nil $K$-theory spectrum $NK(R).$ We also apply the rigidity theorem to define refined versions of negative cyclic homology and periodic cyclic homology. This was announced previously in \cite{E24b}, and certain very interesting examples were computed by Meyer and Wagner in \cite{MW24}. Here we do several computations in characteristic $0,$ in particular showing that in seemingly innocuous situations the answer can be given by an interesting algebra of overconvergent functions.

Theorems & Definitions (191)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Example 1.4
• Example 1.5
• Proposition 1.6