On the notion of geometry over $\F_1$

Details On the notion of geometry over $\F_1$ On the notion of geometry over $\F_1$

2008-09-17

arxiv.org/abs/0809.2926v2

Mathematics

Algebraic Geometry

26 pages

Scientific paper

We refine the notion of variety over the "field with one element" developed by C. Soul\'e 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. Soul\'e 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".

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