Torified varieties and their geometries over {mathbb{F}_1}

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?CConsani 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?CConsani??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.