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 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 Soul\'e and show that both are varieties over 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 1 in the literature so far. Furthermore, we compare Connes-Consani's geometry, Soul\'e's geometry and Deitmar's geometry, and we discuss to what extent Chevalley groups can be realized as group objects over 1 in the given categories.