Global analytic geometry |
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Authors: | Frédéric Paugam |
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Institution: | Université Paris 6, Institut de Mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France |
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Abstract: | Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z, a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and ZX], we define analytic spectra and sheaves of analytic functions on them. |
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Keywords: | Valuation Seminorm Algebraic numbers Analytic functions Number fields |
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