Algebra/Topology Seminar

Speaker: Christian Dahlhausen 

Title: Continuous K-theory and cohomology of rigid spaces

Abstract: Continuous K-theory is a derivative of algebraic K-theory for rigid analytic spaces. In this talk, I will prove three properties for the negative continuous K-theory of a rigid space of dimension d: Weibel vanishing in degrees smaller than -d, analytic homotopy invariance in degrees less or equal -d, and describing the edge group in degree -d as the cohomology in degree d with integral coefficients. These properties are analoguous to correponding properties of negative algebraic K-theory and the proof works by reduction to the algebraic case. The key ingredient is a comparison of Zariski-cohomology and rh-cohomology for certain (relative) Zariski-Riemann spaces. The content of this talk is based on my PhD thesis advised by Moritz Kerz and Georg Tamme, a condensed version can be found on the arXiv (1910.10437).