Partially additive rings and group schemes over ${\mathbb F}_1$

TL;DR

This paper develops a foundational theory of partially additive rings to establish an elementary approach to ${ m F}_1$-geometry, constructing group schemes and projective spaces with applications to counting points over finite fields.

Contribution

It introduces partially additive rings as a basis for ${ m F}_1$-geometry and constructs classical algebraic structures like ${ m GL}_n$ and projective spaces within this framework.

Findings

Constructed a group scheme ${ m GL}_n$ with ${ m F}_1$-points as symmetric groups.

Defined a projective space scheme and counted points over ${ m F}_q$ for q=1 or prime powers.

Explained the significance of the number 1 in ${ m F}_1$ despite having two elements.

Abstract

We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1) we construct a kind of group scheme ${\mathbb GL}_n$ whose value at a commutative ring $R$ is the group of $n\times n$ invertible matrices over $R$ and at ${\mathbb F}_1$ is the $n$-th symmetric group, and (2) we construct a projective space $\mathbb P^n$ as a kind of scheme and count the number of points of ${\mathbb P}^n({\mathbb F}_q)$ for $q=1$ or $q=p^n$ a power of a rational prime, then we explain a reason of number 1 in the subscript of ${\mathbb F}_1$ even though it has two elements.