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December 2018 On topological cyclic homology
Thomas Nikolaus, Peter Scholze
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Acta Math. 221(2): 203-409 (December 2018). DOI: 10.4310/ACTA.2018.v221.n2.a1

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.

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Thomas Nikolaus. Peter Scholze. "On topological cyclic homology." Acta Math. 221 (2) 203 - 409, December 2018. https://doi.org/10.4310/ACTA.2018.v221.n2.a1

Information

Received: 14 October 2017; Revised: 6 September 2018; Published: December 2018
First available in Project Euclid: 19 June 2019

zbMATH: 07009201
MathSciNet: MR3904731
Digital Object Identifier: 10.4310/ACTA.2018.v221.n2.a1

Rights: Copyright © 2018 Institut Mittag-Leffler

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Vol.221 • No. 2 • December 2018
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