Abstract
This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over \({\mathbb Z}\) that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes–Consani and an object in the sense of Soulé and show that both are varieties over \({\mathbb{F}_1}\) in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over \({\mathbb{F}_1}\) in the literature so far. Furthermore, we compare Connes–Consani’s geometry, Soulé’s geometry and Deitmar’s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over \({\mathbb{F}_1}\) in the given categories.
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Acknowledgements
The authors thank all people that participated in the \({\mathbb{F}_1}\)-study seminar, in particular Peter Arndt, Pierre-Emmanuel Chaput, Bram Mesland and Frédéric Paugam for giving interesting lectures at the seminar and participating on stimulating discussions. The authors thank Bas Edixhoven for his help on improving some proofs and Markus Reineke for providing an interesting counter example. The authors thank the Max-Planck Institut für Mathematik in Bonn for support and hospitality and for providing excellent working conditions.
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During the ellaboration of this paper, J. López Peña was supported initially by the Max-Planck-Institut für Mathematik in Bonn, and in the final stages by the EU Marie-Curie fellowship PIEF-GA-2008-221519 at Queen Mary University of London. O. Lorscheid was supported by the Max-Planck-Institut für Mathematik in Bonn.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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López Peña, J., Lorscheid, O. Torified varieties and their geometries over \({\mathbb{F}_1}\) . Math. Z. 267, 605–643 (2011). https://doi.org/10.1007/s00209-009-0638-0
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DOI: https://doi.org/10.1007/s00209-009-0638-0