Continuous K-Theory and Cohomology of Rigid Spaces

TL;DR

This paper reveals a deep connection between continuous K-theory and integral cohomology for rigid spaces, establishing isomorphisms between specific K-groups and cohomology groups based on the space's dimension.

Contribution

It proves an isomorphism between the lowest non-vanishing continuous K-group and the highest non-vanishing cohomology group for rigid spaces, linking K-theory and cohomology in this context.

Findings

Continuous K-groups vanish below negative dimension

Cohomology groups vanish above dimension

Isomorphism between specific K-group and cohomology group

Abstract

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.