Continuous K-Theory and Cohomology of Rigid Spaces

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 a 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.

K-THEORY OF NON-ARCHIMEDEAN RINGS II

“…The main result there was a weak equivalence between continuous and analytic K -theory for regular Noetherian Tate rings admitting a ring of definition satisfying a certain weak resolution of singularities property. Negative continuous K -theory was studied by Dahlhausen in [1].…”

K-theory of non-archimedean rings II

“…The main result there was a weak equivalence between continuous and analytic K-theory for regular noetherian Tate rings admitting a ring of definition satisfying a certain weak resolution of singularities property. Negative continuous K-theory was studied by Dahlhausen in [Dah19].…”