Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry (A Series of Comprehensive Studies in Mathematics; 261)

Review at : http://projecteuclid.org/download/pdf\_1/euclid.bams/1183553480 In the book of BGR (= Bosch-Guntzer-Remmert) a systematic approach to Tate's theory is provided in 415 pages. The book was planned in the late sixties and drafts of a large part of it existed by 1970. It consists of a long part on valuation theory and linear ultrametric analysis that should have been drastically shortened. The parts on affinoid geometry are quite brilliant provided one can appreciate the Bourbaki-type style of presenting mathematics. The word 'affinoid', whose meaning seems to be now very widely known, was suggested by R. Remmert around 1965; it is used to indicate that the affinoid spaces, which are the maximal spectra of topological algebras of finite type over K, are hybrids carrying affine algebraic as well as algebroid features. The prototype of such a space is the closed unit polydisc {x = (x\_l, ... , x\_n) ∈ K^n: |x\_i|) : So eine Illrbeit witb eigentIid) nie rertig, man muli iie fur fertig erfHiren, wenn man nad) 8eit nnb Umftiinben bas moglid)fte get an qat. (@oetqe S. Bosch, U. Güntzer, R. Remmert. Includes Index. Bibliography: P. [416]-420.