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 $\mathbb{F}$.