### SoSe 23 Oberseminar: Selmer K-theory

**Time and place:** Tuesday 14:15-15:45 in M311 and online

Selmer K-theory is a localizing invariant of stable categories introduced by Clausen to give a K-theoretic construction of the Artin map from the idele class group of a number field to its abelianized Galois group. For schemes, Selmer K-theory is closely related to the étale sheafification of algebraic K-theory, and in general it can thus be viewed as a non-commutative extension of the latter.

In this seminar, we will review the definition of Selmer K-theory, which combines insights of Thomason on K(1)-local algebraic K-theory and of Geisser–Hesselholt on topological cyclic homology. We will then discuss applications to étale K-theory following the paper:

- D. Clausen, A. Mathew,
*Hyperdescent and étale K-theory*

For the definitions of Selmer K-theory and of topological cyclic homology we will use:

- D. Clausen,
*A K-theoretic approach to Artin maps* - T. Nikolaus, P. Scholze,
*On topological cyclic homology*

Date | Speaker | Topic |
---|---|---|

18.04 | Niklas Kipp | Introduction |

25.04 | Marc Hoyois | Cyclotomic spectra and topological Hochschild homology |

02.05 | Lucas Piessevaux | Finiteness properties of TC |

09.05 | Marco Volpe | The cyclotomic trace |

16.05 | Ritheesh Krishna Thiruppathi | KU- and K(1)-localization |

23.05 | Niklas Kipp | The Geisser–Levine and Geisser–Hesselholt theorems |

30.05 | — | — |

06.06 | Gabriel Angelini-Knoll | The K-theory of henselian pairs |

13.06 | Christoph Winges | Preliminaries on hypersheaves |

20.06 | Giacomo Bertizzolo | The Nisnevich topos |

27.06 | Niko Naumann | The étale topos |

04.07 | Liam Keenan | Étale descent for TC and KU-localized invariants |

11.07 | Sebastian Wolf | Selmer K-theory and étale K-theory |