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January 2014 The structure of Deitmar schemes, I
Koen Thas
Proc. Japan Acad. Ser. A Math. Sci. 90(1): 21-26 (January 2014). DOI: 10.3792/pjaa.90.21

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.

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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

Published: January 2014
First available in Project Euclid: 6 January 2014

zbMATH: 1329.14009
MathSciNet: MR3161541
Digital Object Identifier: 10.3792/pjaa.90.21

Subjects:
Primary: 14A15
Secondary: 11G25 , 14G15

Keywords: automorphism group , Deitmar scheme , Field with one element , loose graph

Rights: Copyright © 2014 The Japan Academy

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Vol.90 • No. 1 • January 2014
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