Non-additive geometry

We develop a language that makes the analogy between geometry and arithmetic more transparent. In this language there exists a base field IF, 'the field with one element'; there is a fully faithful functor from commutative rings to F-rings; there is the notion of the F-ring of integers of a real or complex prime of a number field K analogous to the p-adic integers, and there is a compactification of SpecO(K); there is a notion of tensor product of F-rings giving the product of IF-schemes; in particular there is the arithmetical surface Spec O-K x Spec O-K, the product taken over F.