# On curves in K-theory and TR

### Jonas McCandless

Max Planck Institute for Mathematics, Bonn, Germany

## 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 with values in the $∞$-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 $∞$-categories, extending the work of Antieau–Nikolaus in the $p$-typical setting. 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.

## Cite this article

Jonas McCandless, On curves in K-theory and TR. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1347