On absolute algebraic geometry the affine case

Abstract

We develop algebraic geometry for general Segal's Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in [3] to define F1-schemes, is naturally a full subcategory of the category of Segal's Γ-rings (equivalently of S-algebras). In this paper we develop the affine case of this general algebraic geometry: one distinctive feature is that the spectrum Spec(A) of an S-algebra is in general a Grothendieck site rather than a point set endowed with a topology. Two striking features of this new geometry are that it is the natural domain for cyclic homology and for homological algebra as in [20], and that new operations, which do not make sense in ordinary algebraic geometry, are here available. For instance, in this new context, the quotient of a ring by a multiplicative subgroup is still an S-algebra to which our general theory applies. Thus the adele class space gives rise naturally to an S-algebra.