Field with one element: Difference between revisions

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In 1957, Jacques Tits introduced the theory of [[building (mathematics)|buildings]], which relate [[algebraic group]]s to [[abstract simplicial complex]]es. One of the assumptions is a non-triviality condition: If the building is an ''n''-dimensional abstract simplicial complex, and if {{nowrap|''k'' < ''n''}}, then every ''k''-simplex of the building must be contained in at least three ''n''-simplices. This is analogous to the condition in classical [[projective geometry]] that a line must contain at least three points. However, there are [[Degeneracy (mathematics)|degenerate]] geometries that satisfy all the conditions to be a projective geometry except that the lines admit only two points. The analogous objects in the theory of buildings are called apartments. Apartments play such a constituent role in the theory of buildings that Tits conjectured the existence of a theory of projective geometry in which the degenerate geometries would have equal standing with the classical ones. This geometry would take place, he said, over a ''field of characteristic one''.<ref>{{harvtxt|Tits|1957}}.</ref> Using this analogy it was possible to describe some of the elementary properties of '''F'''₁, but it was not possible to construct it.

After Tits' initial observations, little progress was made until the early 1990s. In the late 1980s, [[Alexander Smirnov| Alexander Smirnov]] gave a series of talks in which he conjectured that the Riemann hypothesis could be proven by considering the integers as a curve over a field with one element. By 1991, Smirnov had taken some steps towards algebraic geometry over '''F'''₁,<ref name="Smirnov 1992">{{harvtxt|Smirnov|1992}}</ref> introducing extensions of '''F'''₁ and using them to handle the projective line '''P'''¹ over '''F'''₁.<ref name="Smirnov 1992"/> [[Algebraic number]]s were treated as maps to this '''P'''¹, and conjectural approximations to [[Riemann–Hurwitz formula|the Riemann–Hurwitz formula]] for these maps were suggested. These approximations imply very profound assertions like [[abc conjecture|the abc conjecture]]. The extensions of '''F'''₁ later on were denoted as '''F'''_''q'' with ''q'' = 1^''n''. Together with [[Mikhail Kapranov]], Smirnov went on to explore how algebraic and [[number theory|number-theoretic]] constructions in prime characteristic might look in "characteristic one", culminating in an unpublished work released in 1995.<ref>{{harvtxt|Kapranov|Smirnov|1995}}</ref> In 1993, [[Yuri Manin]] gave a series of lectures on [[Riemann zeta function|zeta functions]] where he proposed developing a theory of algebraic geometry over '''F'''₁.<ref>{{harvtxt|Manin|1995}}.</ref> He suggested that zeta functions of [[algebraic variety|varieties]] over '''F'''₁ would have very simple descriptions, and he proposed a relation between the [[algebraic K-theory|K-theory]] of '''F'''₁ and the [[homotopy groups of spheres]]. This inspired several people to attempt to construct explicit theories of '''F'''₁-geometry.

The first published definition of a variety over '''F'''₁ came from [[Christophe Soulé]] in 1999,<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> who constructed it using algebras over the complex numbers and [[functor]]s from [[category (mathematics)|categories]] of certain rings.<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> In 2000, Zhu proposed that '''F'''₁ was the same as '''F'''₂ except that the sum of one and one was one, not zero.<ref>{{harvtxt|Lescot|2009}}.</ref> Deitmar suggested that '''F'''₁ should be found by forgetting the additive structure of a ring and focusing on the multiplication.<ref>{{harvtxt|Deitmar|2005}}.</ref> Toën and Vaquié built on Hakim's theory of relative schemes and defined '''F'''₁ using [[symmetric monoidal category|symmetric monoidal categories]].<ref>{{harvtxt|Toën|Vaquié|2005}}.</ref> Their construction was later shown to be equivalent to Deitmar's by Vezzani.<ref>{{harvtxt|Vezzani|2010}}</ref> [[Nikolai Durov]] constructed '''F'''₁ as a commutative algebraic [[monad (category theory)|monad]].<ref>{{harvtxt|Durov|2008}}.</ref> Borger used [[descent (category theory)|descent]] to construct it from the finite fields and the integers.<ref>{{harvtxt|Borger|2009}}.</ref>