Shai Haron (Technion)
New Foundations for Geometry.

Abstract: 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 develope 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 (!) commutative ring one obtains from the Arithmetical surface Z(x)Z .