On the notion of geometry over 𝔽₁

Connes, Alain; Consani, Caterina

doi:10.1090/S1056-3911-2010-00535-8

On the notion of geometry over $\mathbb {F}_1$

Authors: Alain Connes and Caterina Consani
Journal: J. Algebraic Geom. 20 (2011), 525-557
DOI: https://doi.org/10.1090/S1056-3911-2010-00535-8
Published electronically: December 8, 2010
MathSciNet review: 2786665
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Abstract | References | Additional Information

Abstract: We refine the notion of variety over the “field with one element” developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soulé by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above “field”.

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Additional Information

Alain Connes
Affiliation: Collège de France, 3 rue d’Ulm, Paris F-75005 France; I.H.E.S. and Vanderbilt University
MR Author ID: 51015
Email: alain@connes.org

Caterina Consani
Affiliation: Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218
Email: kc@math.jhu.edu

Received by editor(s): October 13, 2008
Received by editor(s) in revised form: April 3, 2009
Published electronically: December 8, 2010
Additional Notes: The authors are partially supported by the NSF grant DMS-FRG-0652164.