Torified varieties and their geometries over F_1

TL;DR

This paper introduces torified varieties, a new class of schemes decomposable into split tori, and explores their structures over the field with one element, connecting various existing geometries and examples.

Contribution

It defines torified varieties and demonstrates their role as varieties over , linking Connes-Consani, Soule9, and Deitmar's frameworks, and analyzing Chevalley groups over .

Findings

Torified varieties generalize toric and flag varieties.

Constructs gadgets and objects over  from torified varieties.

Provides a unified perspective on geometries over .

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 Soul\'e and show that both are varieties over $\F_1$ 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 $\F_1$ 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 $\F_1$ in the given categories.