论文标题
$ {\ mathbb f} _1 $的部分加性环和组方案
Partially additive rings and group schemes over ${\mathbb F}_1$
论文作者
论文摘要
我们将部分加性环的基本理论开发为$ {\ Mathbb f} _1 $ - 几何的基础。我们的方法是如此具体，以至于对经典代数几何形状的类似物非常简单地建立。 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 $ {\ Mathbb p}^n（{\ Mathbb f} _Q）$的点数对于$ q = 1 $ = 1 $或$ q = p = p = p = p^n $有理素数的幂，那么我们在$ {\ mathbb f} _1 $的下标中解释了一个数字1的原因，即使它有两个元素。
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.
