## Abstract

We describe a construction of the cyclotomic structure on topological Hochschild homology (THH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (TC) and TR-theory: one starting with a ring C_{n}-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory over A in terms of TR over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on TR and TC.

Original language | English (US) |
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Pages (from-to) | 2101-2163 |

Number of pages | 63 |

Journal | Documenta Mathematica |

Volume | 23 |

State | Published - 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Deutsche Mathematiker Vereinigung.

## Keywords

- Adams operations
- Cyclotomic spectrum
- Multiplicative norm
- Topological cyclic homology