On curves in K-theory and TR

Abstract

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the ∞-category of cyclotomic spectra taking values in the ∞-category of cyclotomic spectra with Frobenius lifts, refining a result of Blumberg-Mandell. To that end, we define the notion of an integral topological Cartier module using Barwick's formalism of Mackey functors on orbital ∞-categories, extending the work of Antieau-Nikolaus in the p-typical case. As an application, we show that TR evaluated on a connective E 1 -ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull. Contents 1. Introduction 1 1.1. Statement of results 2 1.2. Methods 3 2. Cyclotomic spectra with Frobenius lifts and TR 6 2.1. The epicyclic category 6 2.2. Spaces with Frobenius lifts 11 2.3. Cyclotomic spectra with Frobenius lifts 15 2.4. Topological restriction homology 18 3. Comparison with genuine TR 20 3.1. Equivariant stable homotopy theory 20 3.2. Topological Cartier modules 24 3.3. Genuine cyclotomic spectra and genuine TR 30 4. Applications to curves on K-theory 34 4.1. Topological Hochschild homology of truncated polynomial algebras 34 4.2. Curves on K-theory 42 References 45