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The structure of Deitmar schemes, I

Koen Thas (UGent)
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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, 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 F-1 (such as combinatorial F-1-projective and 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.
Keywords
Deitmar scheme, Field with one element, loose graph, automorphism group, ZETA-FUNCTIONS, F-1

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Citation

Please use this url to cite or link to this publication:

Chicago
Thas, Koen. 2014. “The Structure of Deitmar Schemes, I.” Proceedings of the Japan Academy Series A-mathematical Sciences 90 (1): 21–26.
APA
Thas, K. (2014). The structure of Deitmar schemes, I. PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, 90(1), 21–26.
Vancouver
1.
Thas K. The structure of Deitmar schemes, I. PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES. 2014;90(1):21–6.
MLA
Thas, Koen. “The Structure of Deitmar Schemes, I.” PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 90.1 (2014): 21–26. Print.
@article{7184092,
  abstract     = {We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F-1) to a so-called {\textacutedbl}loose graph{\textacutedbl} (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 F-1 (such as combinatorial F-1-projective and 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.},
  author       = {Thas, Koen},
  issn         = {0386-2194},
  journal      = {PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES},
  keyword      = {Deitmar scheme,Field with one element,loose graph,automorphism group,ZETA-FUNCTIONS,F-1},
  language     = {eng},
  number       = {1},
  pages        = {21--26},
  title        = {The structure of Deitmar schemes, I},
  volume       = {90},
  year         = {2014},
}

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