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

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 ${\prod }_{n\ge 0}\frac{s-n}{2\pi }$.