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Valuation spectrum and rigid geometry. (Bewertungsspektrum und rigide Geometrie.) (German) Zbl 0806.13001
Regensburger Mathematische Schriften. 23. Regensburg: Fakultät für Mathematik der Universität Regensburg. xi, 303 p. (1993).
The author introduces a new abstract frame work for rigid geometry. The strategy in doing so is the same as the one used by A. Grothendieck when he used schemes to generalize classical algebraic varieties. Much preliminary work is required before the relevant class of spaces, called adic spaces, can be defined. The first chapter contains a discussion of valuation spectra, in the second chapter a class of rings, called \(f\)- adic rings, together with their basic properties is presented. The third chapter starts off with a description of the underlying topological spaces of the affine pieces of adic spaces. These are the spaces of continuous valuations of \(f\)-adic rings. These spaces are endowed with a structure sheaf and some additional valuation theoretic structure to yield affinoid adic spaces and, by gluing these together, adic spaces. Connections with rigid geometry are discussed, examples are given, and some basic properties of adic spaces are investigated.

13A18 Valuations and their generalizations for commutative rings
14G20 Local ground fields in algebraic geometry
32K99 Generalizations of analytic spaces
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13J20 Global topological rings