caa a22 4500 445836091 CHVBK 20180317145330.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00209-009-0638-0 doi (NATIONALLICENCE)springer-10.1007/s00209-009-0638-0 Torified varieties and their geometries over $${\mathbb{F}_1}$$ [Elektronische Daten] [Javier López Peña, Oliver Lorscheid] 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. The Author(s), 2009 López Peña Javier Mathematics Research Centre, Queen Mary University of London, Mile End Road, E1 4NS, London, UK aut Lorscheid Oliver Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany aut Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 605-643 0025-5874 267:3-4<605 2011 267 209 https://doi.org/10.1007/s00209-009-0638-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-009-0638-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- López Peña Javier Mathematics Research Centre, Queen Mary University of London, Mile End Road, E1 4NS, London, UK aut NATIONALLICENCE 700 1- Lorscheid Oliver Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 605-643 0025-5874 267:3-4<605 2011 267 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer