Shai Haran
Non additive geometryAbstract
We give a simple generalization of commutative rings. We show one can develope
algebraic geometry a la Grothendieck using these generalized rings.The initial
object is "the field with one element". The new geometry contains the old one fully
faithfully, but also the "integers" and "residue field" at real and complex primes
of a number field. The arithmetical plane $Z(x)Z$ is not reduced to its diagonal.We
have the module of Kahler differentials satisfying the usual exact sequences, and
the derivatives of integers are non trivial. We have a "zeta mechine" which
associates a meromorfic function to a compact valuation generalized ring : for the
$p$adic integers it gives the $p$ factor of zeta, for the "real integers" it gives
the $\gamma$ factor.
