Supernatural numbers and a new topology on the arithmetic site

TL;DR

This paper introduces a new topology on the arithmetic site’s points, revealing properties like compactness and density that align with the conceptual idea of the mythical ar Spec(Z)/F1, expanding understanding of its structure.

Contribution

It defines a novel topology on the arithmetic site’s points, offering a different perspective that exhibits properties akin to the ar Spec(Z)/F1 space, previously studied mainly via noncommutative geometry.

Findings

The new topology is compact.

It has an uncountable basis of open sets.

Each non-empty open set is dense.

Abstract

In arXiv:1405.4527 Connes and Consani introduced and studied the arithmetic site and showed that the isomorphism classes of points are in canonical bijection with the finite adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^*$. The induced topology of $\mathbb{A}^f_{\mathbb{Q}}$ on this set is trivial, whence this space is usually studied via noncommutative geometry. However, we can define another topology on this set of points, which shares several properties one might expect of the mythical object $\overline{\mathbf{Spec}(\mathbb{Z})}/\mathbb{F}_1$: it is compact, has an uncountable basis of opens, each non-empty open being dense, and it satisfies the $T_1$ separation property for incomparable points.