The structure of Deitmar schemes, I

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Koen Thas¹ 

[1] Ghent University, Department of Mathematics

关键词: Field with one element;    Deitmar scheme;    loose graph;    automorphism group.;   

DOI  :  10.3792/pjaa.90.21

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $\mathbf{F}_{1}$) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and it also appears that known realizations of objects over $\mathbf{F}_{1}$ (such as combinatorial $\mathbf{F}_{1}$-projective and $\mathbf{F}_{1}$-affine spaces) exactly depict the loose graph which corresponds to the associated Deitmar scheme. This idea is then conjecturally generalized so as to describe all Deitmar schemes in a similar synthetic manner.

【 授权许可】

Unknown