New foundations for geometry

Abstract

We shall describe a simple generalization of commutative rings. The category GR of such "rings", contains the ordinary commutative rings (fully faithfully), but also the "integers" and "residue field" at a real or complex place of a field ; the "field with one element" (the initial object of GR ); the "arithmetical surface" ( the sum in the category GR of the integers with them self: Z(x)Z ) . We shall show that this geometry "see" the real and complex places of a number field (there is an Ostrowski theorem that the valuation sub-GR of a number field correspond to the finite and infinite primes; there is a compactification of the spectrum of the integers ). One can develop algebraic geometry using GR following Grothendieck paradigm . Quillen's homotopical algebra replaces homological algebra . There is an analogue of Quillen's cotangent bundle - in particular there is a theory of non-additive derivations, with modules of Kahler differentials which satisfy all the usual exact sequences - we compute explicitly the differentials of the integers Z over the initial object F1 . Finally we associate with any topological compact valuation GR a meromorphic function - its "zeta" : for the p-adic integers we get the p-local factor of zeta, for the "real integers" we get the gamma factor.