Torified varieties and their geometries over F1

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 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 F1 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 F1 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 F1 in the given categories. © 2009 The Author(s).