On topological cyclic homology

Thomas Nikolaus Universität Münster Peter Scholze Universität Bonn

K-Theory and Homology mathscidoc:1911.43019

Acta Mathematica, 221, (2), 203 – 409
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 ∞-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum X with S1-action (in the most naive sense) together with S1-equivariant maps φp:X→XtCp for all primes p. Here, XtCp=cofib(Nm:XhCp→XhCp) 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 φp:X→XtCp 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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  title={On topological cyclic homology},
  author={Thomas Nikolaus, and Peter Scholze},
  booktitle={Acta Mathematica},
  pages={203 – 409},
Thomas Nikolaus, and Peter Scholze. On topological cyclic homology. Vol. 221. In Acta Mathematica. pp.203 – 409. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191126162658947830530.
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