Field with one element: Difference between revisions

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{{Use dmy dates|date=September 2020}}

In [[mathematics]], the '''field with one element''' is a suggestive name for an object that should behave similarly to a [[finite field]] with a single element, if such a field could exist. This object is denoted '''F'''₁, or, in a French–English pun, '''F'''_un.<ref>"[[wikt:un#French|un]]" is French for "one", and [[wikt:fun|fun]] is a playful English word. For examples of this notation, see, e.g. {{harvtxt|Le Bruyn|2009}}, or the links by Le Bruyn, Connes, and Consani.</ref> The name "field with one element" and the notation '''F'''₁ are only suggestive, as there is no field with one element in classical [[abstract algebra]]. Instead, '''F'''₁ refers to the idea that there should be a way to replace sets[[set (mathematics)|set]]s and operations[[Operation (mathematics)|operation]]s, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of '''F'''₁ have been proposed, but it is not clear which, if any, of them give '''F'''₁ all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose [[characteristic (algebra)|characteristic]] is one.

Most proposed theories of '''F'''₁ replace abstract algebra entirely. Mathematical objects such as [[vector space]]s and [[polynomial ring]]s can be carried over into these new theories by mimicking their abstract properties. This allows the development of [[commutative algebra]] and [[algebraic geometry]] on new foundations. One of the defining features of theories of '''F'''₁ is that these new foundations allow more objects than classical abstract algebra, one of which behaves like a field of characteristic one.

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==History==

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 whichthat 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, [[Smirnov Alexander Leonidovich| 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 functors[[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>

[[Alain Connes]] and [[Caterina Consani]] developed both Soulé and Deitmar's notions by "gluing" the category of multiplicative monoids[[monoid]]s 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- and 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>

Oliver Lorscheid, along with others, has recently achieved Tits' original aim of describing Chevalley groups over '''F'''₁ by introducing objects called blueprints, which are a simultaneous generalisation of both [[semiring]]s and monoids.<ref name=":0"/><ref>{{harv|Lorscheid|2018b}}</ref> These are used to define so-called "blue schemes", one of which is Spec '''F'''₁.<ref>{{harvtxt|Lorscheid|2016}}</ref> Lorscheid's ideas depart somewhat from other ideas of groups over '''F'''₁, in that the '''F'''₁-scheme is not itself the Weyl group of its base extension to normal schemes. Lorscheid first defines the Tits category, a full subcategory of the category of blue schemes, and defines the "Weyl extension", a functor from the Tits category to '''Set'''. A Tits-WeylTits–Weyl model of an algebraic group <math>\mathcal{G}</math> is a blue scheme ''G'' with a group operation whichthat is a morphism in the Tits category, whose base extension is <math>\mathcal{G}</math> and whose Weyl extension is isomorphic to the Weyl group of <math>\mathcal{G}.</math>

'''F'''₁-geometry has been linked to [[tropical geometry]], via the fact that semirings (in particular, tropical semirings) arise as quotients of some monoid semiring '''N'''[''A''] of finite formal sums of elements of a monoid ''A'', which is itself an '''F'''₁-algebra. This connection is made explicit by Lorscheid's use of blueprints.<ref>{{harvtxt|Lorscheid|2015}}</ref> The Giansiracusa brothers have constructed a tropical scheme theory, for which their category of tropical schemes is equivalent to the category of Toën-VaquiéToën–Vaquié '''F'''₁-schemes.<ref>{{harvtxt|Giansiracusa|Giansiracusa|2016}}</ref> This category embeds [[faithful functor|faithfully]], but not [[full functor|fully]], into the category of blue schemes, and is a [[full subcategory]] of the category of Durov schemes.

==Motivations==

===Algebraic number theory===

One motivation for '''F'''₁ comes from [[algebraic number theory]]. [[Andrew Weil|Weil]]'s proof of the [[Riemann hypothesis for curves over finite fields]] starts with a curve ''C'' over a finite field ''k'', which comes equipped with a [[Function field of an algebraic variety|function field]] ''F'', which is a [[field extension]] of ''k''. Each such function field gives rise to a [[Hasse–Weil zeta function]] {{math|ζ_''F''}}, and the Riemann hypothesis for finite fields determines the zeroes of {{math|ζ_''F''}}. Weil's proof then uses various geometric properties of ''C'' to study {{math|ζ_''F''}}.

The field of rational numbers '''Q''' is linked in a similar way to the [[Riemann zeta function]], but '''Q''' is not the function field of a variety. Instead, '''Q''' is the function field of the [[scheme (mathematics)|scheme]] {{math|Spec '''Z'''}}. This is a one-dimensional scheme (a.k.a.also known as) an [[algebraic curve]]), and so there should be some "base field" whichthat this curve lies over, of which '''Q''' would be a [[field extension]] (in the same way that ''C'' is a curve over ''k'', and ''F'' is an extension of ''k''). The hope of '''F'''₁-geometry is that a suitable object '''F'''₁ could play the role of this base field, which would allow for a proof of the [[Riemann hypothesis]] by mimicking Weil's proof with '''F'''₁ in place of ''k''.

===Arakelov geometry===

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===Monoid schemes===

The ''spectrum'' of a monoid {{math |''A'',}} denoted {{math |Spec ''A'',}} is the set of [[prime idealsideal]]s of {{math |''A''.}} The spectrum of a monoid can be given a [[Zariski topology]], by defining [[basis (topology)|basic]] [[open setsset]]s

:<math>U_h = \{\mathfrak{p}\in\text{Spec}A:h\notin\mathfrak{p}\},</math>

for each {{math |''h''}} in {{math |''A''.}} A ''monoidal space'' is a topological space along with a [[sheaf (mathematics)|sheaf]] of multiplicative monoids called the ''structure sheaf''. An ''affine monoid scheme'' is a monoidal space which is isomorphic to the spectrum of a monoid, and a '''monoid scheme''' is a sheaf of monoids which has an open cover by affine monoid schemes.

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This construction achieves many of the desired properties of '''F'''₁-geometry: {{math |Spec '''F'''₁}} consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative rings. Furthermore, this theory satisfies the combinatorial properties expected of '''F'''₁ mentioned in previous sections; for instance, projective space over {{math |'''F'''₁}} of dimension {{math|''n''}} as a monoid scheme is identical to an apartment of projective space over {{math |'''F'''_''q''}} of dimension {{math|''n''}} when described as a building.

However, monoid schemes do not fulfill all of the expected properties of a theory of '''F'''₁-geometry, as the only varieties whichthat have monoid scheme analogues are [[toric variety|toric varieties]].<ref>{{harvtxt|Deitmar|2006}}</ref> More precisely, if {{math |''X''}} is a monoid scheme whose base extension is a [[Flat morphism|flat]], [[Glossary of algebraic geometry#S|separated]], [[Connected space|connected]] scheme of [[Finite morphism#Morphisms of finite type|finite type]], then the base extension of {{math |''X''}} is a toric variety. Other notions of '''F'''₁-geometry, such as that of Connes–Consani,<ref>{{harvtxt|Connes|Consani|2010}}</ref> build on this model to describe '''F'''₁-varieties which are not toric.

==Field extensions==