## Acta Mathematica

- Acta Math.
- Volume 221, Number 2 (2018), 203-409.

### On topological cyclic homology

Thomas Nikolaus and Peter Scholze

#### Abstract

Topological cyclic homology is a refinement of Connes–Tsygan’s cyclic homology which was introduced by Bökstedt–Hsiang–Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas–Goodwillie–McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum.

The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p : X \to X^{t C_p}$ for all primes $p$. Here, $X^{t C_p} = \mathrm{cofib}(\mathrm{Nm} : X^{h C_p} \to X^{h C_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology.

In order to construct the maps $\varphi_p : X \to X^{t C_p}$ in the example of topological Hochschild homology, we introduce and study Tate-diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular, we prove a version of the Segal conjecture for the Tate-diagonals and relate these Frobenius homomorphisms to power operations.

#### Article information

**Source**

Acta Math., Volume 221, Number 2 (2018), 203-409.

**Dates**

Received: 14 October 2017

Revised: 6 September 2018

First available in Project Euclid: 19 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.acta/1560967045

**Digital Object Identifier**

doi:10.4310/ACTA.2018.v221.n2.a1

**Mathematical Reviews number (MathSciNet)**

MR3904731

**Zentralblatt MATH identifier**

07009201

#### Citation

Nikolaus, Thomas; Scholze, Peter. On topological cyclic homology. Acta Math. 221 (2018), no. 2, 203--409. doi:10.4310/ACTA.2018.v221.n2.a1. https://projecteuclid.org/euclid.acta/1560967045