Abstract
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.
Citation
Koen Thas. "The structure of Deitmar schemes, I." Proc. Japan Acad. Ser. A Math. Sci. 90 (1) 21 - 26, January 2014. https://doi.org/10.3792/pjaa.90.21
Information