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 $\mathbb{F}$ , ‘the field with one element’; there is a fully faithful functor from commutative rings to $\mathbb{F}$ -rings; there is the notion of the $\mathbb{F}$ -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 $\operatorname{Spec}O_K$ ; there is a notion of tensor product of $\mathbb{F}$ -rings giving the product of $\mathbb{F}$ -schemes; in particular there is the arithmetical surface $\operatorname{Spec} O_K\times\operatorname{Spec} O_K$ , the product taken over $\mathbb{F}$ .