The coordinate biring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$

TL;DR

This paper introduces a novel approach to defining $ ext{Spec}( ext{Z})/ ext{F}_1$ using integral bi-rings with descent data, linking algebraic geometry over $ ext{F}_1$ to recursive sequences and noncommutative moduli spaces.

Contribution

It proposes a new definition of $ ext{F}_1$-algebras as integral bi-rings and constructs a related noncommutative moduli space with a specific motive.

Findings

Coordinate bi-ring is the co-ring of integral linear recursive sequences.

The noncommutative moduli space is defined over $ ext{F}_1$.

The motive of the moduli space is explicitly computed.

Abstract

We propose to define $\mathbb{F}_1$-algebras as integral bi-rings with the co-ring structure being the descent data from $\mathbb{Z}$ to $\mathbb{F}_1$. The coordinate bi-ring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ is then the co-ring of integral linear recursive sequences equipped with the Hadamard product. We associate a noncommutative moduli space to this setting, show that it is defined over $\mathbb{F}_1$, and has motive $\prod_{n \geq 0} \frac{s-n}{2 \pi}$.