Proceedings of the Japan Academy, Series A, Mathematical Sciences

The structure of Deitmar schemes, I

Koen Thas

Full-text: Open access

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


Export citation

References

  • A. Deitmar, Schemes over $\mathbf{F}_{1}$, in Number fields and function fields,–-,two parallel worlds, Progr. Math., 239, Birkhäuser Boston, Boston, MA, 2005, pp. 87–100.
  • A. Deitmar, Remarks on zeta functions and $K$-theory over $\mathbf{F}_{1}$, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141–146.
  • J. de Groot, Groups represented by homeomorphism groups, Math. Ann. 138 (1959), 80–102.
  • R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6 (1939), 239–250.
  • H. Izbicki, Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften, Monatsh. Math. 63 (1959), 298–301.
  • N. Kurokawa, Zeta functions over $\mathbf{F}_{1}$, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 180–184.
  • J. López Peña and O. Lorscheid, Mapping $\mathbf{F}_{1}$-land: an overview of geometries over the field with one element, in Noncommutative geometry, arithmetic, and related topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, pp. 241–265.
  • O. Lorscheid, A blueprinted view on $\mathbf{F}_{1}$-geometry, in Absolute Arithmetic and $\mathbf{F}_{1}$-Geometry. (Submitted).
  • K. Thas, Notes on $\mathbf{F}_{1}$, I, Unpublished notes, 2012.
  • K. Thas, The Weyl functor,–-,Introduction to Absolute Arithmetic, in Absolute Arithmetic and $\mathbf{F}_{1}$-Geometry. (Submitted).
  • K. Thas, The combinatorial-motivic nature of $\mathbf{F}_{1}$-schemes, in Absolute Arithmetic and $\mathbf{F}_{1}$-Geometry. (Submitted).
  • K. Thas (ed.), Absolute Arithmetic and $\mathbf{F}_{1}$-Geometry. (Submitted).
  • J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain, 1957, pp. 261–289.