On curves in K-theory and TR

TL;DR

This paper advances the understanding of TR in algebraic K-theory by showing its corepresentability via topological Hochschild homology, introducing integral topological Cartier modules, and relating TR to spectra of algebraic curves.

Contribution

It proves TR's corepresentability by reduced topological Hochschild homology, defines integral topological Cartier modules, and links TR to spectra of algebraic curves on K-theory, extending prior results.

Findings

TR is corepresentable by reduced topological Hochschild homology of $ extbf{S}[t]$

Introduction of integral topological Cartier modules

Description of TR on connective $ extbf{E}_1$-rings in terms of algebraic curves

Abstract

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line $\mathbf{S}[t]$ as a functor defined on the $\infty$-category of cyclotomic spectra with values in the $\infty$-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital $\infty$-categories, extending the work of Antieau-Nikolaus in the $p$-typical setting. As an application, we show that TR evaluated on a connective $\mathbf{E}_1$-ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull.