New Foundations for Geometry: Two Non-Additive Languages for Arithmetical Geometry

We give two simple generalizations of commutative rings. They form (co)-complete categories, that contain commutative (semi-) rings (e.g. with the usual multiplication ). But they also contains the "integers" (and ), and the "residue fields" (and ), of the real (and complex) numbers. Here is the collection of unit balls, and is the collection of spheres augmented with a . The initial object is "the field with one element" .One generalization, - the "commutative generalized rings", is an axiomatization of finitely generated free modules over a commutative ring, together with the operations of multiplication and contraction. This is the more geometric language: for any we associate its (symmetric) spectrum, , a compact Zariski space, with a sheaf of over it. By glueing such spectra we get generalized schemes , a full sub-category of the locally-generalized-ringed-spaces. For a number field , with the ring of integers , the compatification of is a pro-object , and its points are the valuation-sub- of : .For , we have a (co)-complete abelian category of - modules with enough injectives and projectives. For in , we obtain the - module of Kähler differentials , satisfying all the usual properties. We compute the universal derivation .All these remain true for the second generalization - the "commutative with involution", the axiomatization of the category of finitely generated free -modules with -linear maps, and the operations of composition,direct sum, and taking transpose.This is the more "linear", or K-theoretic language: for , we have its algebraic K-theory spectum: , and for we obtain the sphere spectrum .For a compact valuation we associate a "zeta" function, so that we obtain the usual factor for the p-adic integers , while we get for the real integers .For , we define the category of vector bundles over , by a certain completion of the categories of vector bundles on the finite layers . For a number field , the isomorphism classes of rank vector bundles over are in natural bijection withwhere (resp. ) for real (resp. complex) place of . E.g. for : , and for : .We have the following "commutative" diagram of adjunctions:where is the left adjoint of the forgetfull functor and .We describe the ordinary commutative (semi)- ring associated by the right adjoint functor to the - fold tensor product (resp. ).Its elements are (non-uniquely) represented as , where are finite rooted trees, with maps , and is a bijection of their leaves , and for we have in addition signs.