On the notion of geometry over $\F_1$

Abstract

We refine the notion of variety over the "field with one element" developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soulé by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above "field".

26 pages

References in corpus (2)

• New Approach to Arakelov Geometry
• Les variétés sur le corps à un élément

Cited by in corpus (8)

• Cyclotomy and analytic geometry over F_1
• On the Hall algebra of semigroup representations over F_1
• Algebraic groups over the field with one element
• Representations of Quivers over F1
• On the Hall algebra of coherent sheaves on P^1 over F_1
• Functional equations for zeta functions of ${\mathbb{F}}_1$-schemes
• (non)commutative f-un geometry
• Cyclotomy and endomotives