We shall give a simple generalization of commutative rings. The
category GR of such generalized rings contains ordinary commutative
rings (fully, faithfully), but also the "integers" and the "residue
field" at a real or complex place of a number field ; the "field with
one element" F1 (the initial object of GR) ; the "Arithmetical
Surface" (the categorical sum of the integers Z with them self).
We shall show this geometry sees the real and complex places of a number
field K : the valuation sub GR of K correspond to the finite and
infinite primes of K, and there is a compactification of the spectrum
of the integers of K.
One can develop algebraic geometry using generalized rings following
Grothendieck's paradigm, with Quillen's homotopical algebra replacing
homological algebra.
There is a theory of differentials which satisfy all the usual properties,
as well as an analogue of Quillen's cotangent complex. We compute the
differentials of the integers Z over F1.
We associate with any compact topological valuation generalized ring its
zeta function so that for the p-adic integers we get the usual factor of zeta,
while for the real integers we get the gamma factor. Finally we describe
the remarkable ordinary ring one obtains from the Arithmetical surface.