Global analytic geometry

Details Global analytic geometry Global analytic geometry

2008-03-03

arxiv.org/abs/0803.0148v2

Mathematics

Algebraic Geometry

39 pages; in Journal of Number Theory 2009

Scientific paper

Motivated by the well-known lack of archimedean information in algebraic geometry, we define a new type of valuation of a ring that combines the notion of Krull valuation with that of 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 space.After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.

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