The structure of Deitmar schemes. I. (English) Zbl 1329.14009

Summary: We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, \(\mathbb{F}_{1}\)) 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 \(\mathbb{F}_{1}\) (such as combinatorial \(\mathbb{F}_{1}\)-projective and \(\mathbb{F}_{1}\)-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.


14A15 Schemes and morphisms
14G15 Finite ground fields in algebraic geometry
11G25 Varieties over finite and local fields
14G20 Local ground fields in algebraic geometry
Full Text: DOI Euclid


[1] 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. · Zbl 1098.14003 · doi:10.1007/0-8176-4447-4_6
[2] 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. · Zbl 1173.14004 · doi:10.3792/pjaa.82.141
[3] J. de Groot, Groups represented by homeomorphism groups, Math. Ann. 138 (1959), 80-102. · Zbl 0087.37802 · doi:10.1007/BF01369667
[4] R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6 (1939), 239-250. · Zbl 0020.07804
[5] H. Izbicki, Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften, Monatsh. Math. 63 (1959), 298-301. · Zbl 0086.16103 · doi:10.1007/BF01295203
[6] N. Kurokawa, Zeta functions over \(\mathbf{F}_{1}\), Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 180-184. · Zbl 1141.11316 · doi:10.3792/pjaa.81.180
[7] 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. · Zbl 1271.14003
[8] O. Lorscheid, A blueprinted view on \(\mathbf{F}_{1}\)-geometry, in Absolute Arithmetic and \(\mathbf{F}_{1}\)-Geometry . (Submitted).
[9] K. Thas, Notes on \(\mathbf{F}_{1}\), I, Unpublished notes, 2012.
[10] K. Thas, The Weyl functor,–,Introduction to Absolute Arithmetic, in Absolute Arithmetic and \(\mathbf{F}_{1}\)-Geometry . (Submitted). · Zbl 0435.53042
[11] K. Thas, The combinatorial-motivic nature of \(\mathbf{F}_{1}\)-schemes, in Absolute Arithmetic and \(\mathbf{F}_{1}\)-Geometry . (Submitted). · Zbl 0435.53042
[12] K. Thas (ed.), Absolute Arithmetic and \(\mathbf{F}_{1}\)-Geometry . (Submitted).
[13] 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. · Zbl 0084.15902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.