  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. 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.

Algebraic Geometry;Number Theory