Non-additive geometry

Abstract

We develop a language that makes the analogy between geometry and arithmetic more transparent. In this language there exists a base field double-struck F sign, 'the field with one element'; there is a fully faithful functor from commutative rings to double-struck F sign-rings; there is the notion of the double-struck F sign-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 Spec OK; there is a notion of tensor product of double-struck F sign-rings giving the product of double-struck F sign-schemes; in particular there is the arithmetical surface Spec OK × Spec OK, the product taken over F. © Foundation Compositio Mathematica. 2007.