Ebook: Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry

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| <= 1} which is the spectrum of maximal ideals of the K-algebra of strictly convergent power series in the variables x_l, ... , x_n. Here K is also assumed to be algebraically closed. The book of BGR gives a survey of the main results of the research carried out between 1965 and 1970 by a group of persons in Gottingen and Munster, led by Grauert and Remmert. Their interest was concentrated on the abstract concepts. Among the more important results were finiteness theorems for the reduction functor and the functor of power-bounded elements, the proper mapping theorem, and the characterization of the locally closed immersions. Rigid analytic varieties are defined in BGR using Grothendieck topologies given by systems of admissible open subsets and admissible coverings; they have to satisfy the condition that there exists an admissible covering of the entire space by affinoid subdomains. This approach is due to R. Kiehl. Unfortunately no attempt is made to relate this concept to Tate's h-structures, which are defined by selecting morphisms. In some respects the book of BGR does not carry the subject very far. It makes no mention of differentials, derivations, or vector fields. There are not enough interesting examples of rigid analytic varieties. There are almost no indications of the more exciting developments of recent years. The application to elliptic curves on the last pages is too meager.

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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) E K^n: |x_i| <= 1} which is the spectrum of maximal ideals of the K-algebra of strictly convergent power series in the variables x_l, ... , x_n. Here K is also assumed to be algebraically closed. The book of BGR gives a survey of the main results of the research carried out between 1965 and 1970 by a group of persons in Gottingen and Munster, led by Grauert and Remmert. Their interest was concentrated on the abstract concepts. Among the more important results were finiteness theorems for the reduction functor and the functor of power-bounded elements, the proper mapping theorem, and the characterization of the locally closed immersions. Rigid analytic varieties are defined in BGR using Grothendieck topologies given by systems of admissible open subsets and admissible coverings; they have to satisfy the condition that there exists an admissible covering of the entire space by affinoid subdomains. This approach is due to R. Kiehl. Unfortunately no attempt is made to relate this concept to Tate's h-structures, which are defined by selecting morphisms. In some respects the book of BGR does not carry the subject very far. It makes no mention of differentials, derivations, or vector fields. There are not enough interesting examples of rigid analytic varieties. There are almost no indications of the more exciting developments of recent years. The application to elliptic curves on the last pages is too meager.