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Journal Article

#### On topological cyclic homology

##### External Ressource

https://dx.doi.org/10.4310/ACTA.2018.v221.n2.a1

(Publisher version)

##### Fulltext (public)

arXiv:1707.01799.pdf

(Preprint), 2MB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Nikolaus, T., & Scholze, P. (2018). On topological cyclic homology.*
Acta Mathematica,* *221*(2), 203-409. doi:10.4310/ACTA.2018.v221.n2.a1.

Cite as: http://hdl.handle.net/21.11116/0000-0008-00AA-8

##### Abstract

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic
homology which was introduced by B\"okstedt--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 $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here
$X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ 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 $\varphi_p: X\to X^{tC_p}$ 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.