Torified varieties and their geometries over F₁

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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 Soulé 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, Soulé'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.