*** Please note the LOCATION ***

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 ring one obtains from the Arithmetical surface.

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 ring one obtains from the Arithmetical surface.

## Date:

Thu, 09/06/2016 - 12:00 to 13:15

## Location:

Manchester Building (Ross 63), Jerusalem, Israel