Abstract
Let X be a rigid analytic space and assume that X admits a formal covering inducing on X the structure of a formal scheme. Then, dependent on such a formal structure one associates to X the “reduction”\(\tilde X\) which is a scheme of locally finite type over the residue field corresponding to the ground field in question. In this paper the cohomology groups of X are compared with the cohomology groups of\(\tilde X\) and a dimension formula is proved. As a consequence it is shown that X is affinoid\(\tilde X\) is affine and that an analytic map ϕ:X→Y which is compatible with the formal structures on X and Y is finite if and only if\(\tilde \varphi :\tilde X \to \tilde Y\) is finite.
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Bosch, S. Zur Kohomologietheorie rigid analytischer Räume. Manuscripta Math 20, 1–27 (1977). https://doi.org/10.1007/BF01181238
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DOI: https://doi.org/10.1007/BF01181238