New foundations for geometry

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.