Partially additive rings and group schemes over \mathbb F₁

2022/06/13 by Okuyama, Shingo
#14A23(Primary) #14L17(Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.06084

Abstract

We develop an elementary theory of partially additive rings as a foundation of \mathbb F₁-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× n invertible matrices over R and at \mathbb F₁ is the n-th symmetric group, and (2) we construct a projective space \mathbb Pⁿ as a kind of scheme and count the number of points of \mathbb Pⁿ(\mathbb F_q) for q=1 or q=pⁿ a power of a rational prime, then we explain a reason of number 1 in the subscript of \mathbb F₁ even though it has two elements.

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