Proceedings of the Japan Academy, Series A, Mathematical Sciences

The structure of Deitmar schemes, I

Koen Thas

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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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 1 (2014), 21-26.

Dates
First available in Project Euclid: 6 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.pja/1389017411

Digital Object Identifier
doi:10.3792/pjaa.90.21

Mathematical Reviews number (MathSciNet)
MR3161541

Zentralblatt MATH identifier
1329.14009

Subjects
Primary: 14A15: Schemes and morphisms
Secondary: 14G15: Finite ground fields 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

Keywords
Field with one element Deitmar scheme loose graph automorphism group

Citation

Thas, Koen. The structure of Deitmar schemes, I. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 1, 21--26. doi:10.3792/pjaa.90.21. https://projecteuclid.org/euclid.pja/1389017411


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References

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