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

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 ${\mathbb{G}L}_n$ 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.