# Oberseminar Winter Term 2019/20

**Speaker: **Christian Dahlhausen (Zürich)

**Date, time & place:** January 23, 2020, 4:45 pm, WSC-N-U-3.05

**Title: **Continuous K-theory and K-theory of Zariski-Riemann spaces

**Abstract:** Continuous K-theory is a variant of algebraic K-theory for rigid spaces (nonarchimedean analytic spaces). First, I will state Morrow's definition and some basic results by Kerz-Saito-Tamme which are used in the remainder. In the spirit of Raynaud's description of rigid spaces it is fruitful to study the continuous K-theory of a rigid space in terms of the K-theory of all its integral models; this yields us to study the K-theory of Zariski-Riemann spaces which behaves similar to the K-theory of regular schemes (due to a result by Kerz-Strunk based on Raynaud-Gruson's "platification par eclatement"). As an application I will relate the bottom continuous K-group of a rigid space (over a complete discretely valued field) to its top cohomology group. This is the main result of my PhD thesis advised by Moritz Kerz and Georg Tamme, a condensed version can be found on the arXiv (1910.10437). In the end, I will mention some further progress towards more general base fields.