Field with one element: Difference between revisions

Revision as of 07:49, 7 February 2020

Revision as of 17:04, 22 February 2020

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By 1991, Alexander Smirnov had taken some steps towards algebraic geometry over '''F'''₁.<ref>{{harvtxt|Smirnov|1992}}</ref> He introduced extensions of '''F'''₁ and used them to handle the projective line '''P'''¹ over '''F'''₁. [[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<ref>{{harvtxt|Kapranov|Smirnov|1995}}</ref> as '''F'''_''q'' with ''q'' = 1^''n''.

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 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 '''F'''₁. 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> Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings.<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> Borger used [[descent (category theory)|descent]] to construct it from the finite fields and the integers.<ref>{{harvtxt|Borger|2009}}.</ref>

[[Alain Connes]] and [[Caterina Consani]] developed both Soulé and Deitmar's notions by "gluing" the category of multiplicative monoids and the category of rings to create a new category <math>\mathfrak{M}\mathfrak{R},</math> then defining '''F'''₁-schemes to be a particular kind of representable functor on <math>\mathfrak{M}\mathfrak{R}.</math><ref>{{harvtxt|Connes|Consani|2010}}.</ref> Using this, they managed to provide a notion of several number-theoretic constructions over '''F'''₁ such as motives and field extensions, as well as constructing [[Groups of Lie type#Chevalley groups|Chevalley groups]] over '''F'''_1². Along with [[Matilde Marcolli]], Connes-Consani have also connected '''F'''₁ with [[noncommutative geometry]].<ref>{{harvtxt|Connes|Consani|Marcolli|2009}}</ref> It has also been suggested to have connections to the [[unique games conjecture]] in [[computational complexity theory]].<ref>{{citation|url=https://gilkalai.wordpress.com/2018/01/10/subhash-khot-dor-minzer-and-muli-safra-proved-the-2-to-2-games-conjecture/|title=Subhash Khot, Dor Minzer and Muli Safra proved the 2-to-2 Games Conjecture|work=Combinatorics and more|first=Gil|last=Kalai|authorlink=Gil Kalai|date=2018-01-10}}</ref>