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

Continuous K-Theory and Cohomology of Rigid Spaces