Localization and Affine Schemes over ${\mathbb{F}}_1$

Abstract

We develop the basic notions of commutative algebra and algebraic geometry over the field with one element ${\mathbb{F}}_1$, working within the Connes-Consani framework, which models ${\mathbb{F}}_1$-algebras as monoid objects in the category of $\hat{I}“$-sets. In this setting, ${\mathbb{F}}_1$-algebras generalize commutative rings by encoding the algebraic structure functorially, using machinery originating in homotopy theory. Our main contribution is a theory of localization for ${\mathbb{F}}_1$-algebras and the construction of prime spectrum $\Spec A$ for a commutative ${\mathbb{F}}_1$-algebra $A$. We then prove that $\hat{I}“(X,{\mathcal{O}}_X)=A$ for any absolute affine scheme $X=\Spec A$ and establish an anti-equivalence between the category of commutative ${\mathbb{F}}_1$-algebras and the category of absolute affine schemes.

22 pages