Field with one element: Difference between revisions

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[[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>

Lorscheid, along with others, has recently achieved Tit's 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>{{harvtxt|Lorscheid|2018a}}</ref><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-Weyl model of an algebraic group <math>\mathcal{G}</math> is a blue scheme ''G'' with a group operation which 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-Vaquié '''F'''₁-schemes.<ref>{{harvtxt|Giansiracusa|Giansiracusa|2016}}</ref> This category embeds faithfully, but not fully, into the category of blue schemes, and is a full subcategory of the category of Durov schemes.

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===Monoids===

A '''multiplicative monoid''' is a monoid {{math |''A''}} which also contains an [[absorbing element]] 0 (distinct from the identity 1 of the monoid), such that {{math |0''a'' {{=}} 0}} for every {{math |''a''}} in the monoid {{math |''A''.}} The field with one element is then defined to be {{math |'''F'''₁ {{=}} {0,1},}} the multiplicative monoid of the field with two elements, which is [[initial object|initial]] in the category of multiplicative monoids. A '''monoid ideal''' in a monoid {{math |''A''}} is a subset {{math |''I''}} which is multiplicatively closed, contains 0, and such that {{math |''IA'' {{=}} {''ra'' : ''r''∈''I'', ''a''∈''A''} {{=}} ''I''.}} Such an ideal is '''prime''' if <math>A\setminus I</math> is multiplicatively closed and contains 1.

For monoids {{math |''A''}} and {{math |''B'',}} a '''monoid homomorphism''' is a function {{math |''f'' : ''A'' → ''B''}} such that;

* {{math |''f''(0) {{=}} 0}};

* {{math |''f''(1) {{=}} 1,}} and

* {{math |''f''(''ab'') {{=}} ''f''(''a'')''f''(''b'')}} for every {{math |''a''}} and {{math |''b''}} in {{math |''A''.}}

===Monoid schemes===

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Monoid schemes can be turned into ring-theoretic schemes by means of a '''base extension''' [[functor]] <math>-\otimes_{\mathbf{F}_1}\mathbf{Z}</math> which sends the monoid ''A'' to the '''Z'''-module (i.e. ring) <math>\mathbf{Z}[A]/\langle 0_A\rangle,</math> and a monoid homomorphism {{math |''f'' : ''A'' → ''B''}} extends to a ring homomorphism <math>f_{\mathbf{Z}}:A\otimes_{\mathbf{F}_1}\mathbf{Z}\to B\otimes_{\mathbf{F}_1}\mathbf{Z}</math> which is linear as a '''Z'''-module homomorphism. The base extension of an affine monoid scheme is defined via the formula

:<math>\operatorname{Spec}(A)\times_{\operatorname{Spec}(\mathbf{F}_1)}\operatorname{Spec}(\mathbf{Z})=\operatorname{Spec}\big( A\otimes_{\mathbf{F}_1}\mathbf{Z}\big),</math>

which in turn defines the base extension of a general monoid scheme.

===Consequences===

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* {{citation | editor1-last=Consani | editor1-first=Caterina | editor2-last=Connes | editor2-first=Alain | editor2-link=Alain Connes | title=Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23–26, 2009 | ___location=Baltimore, MD | publisher=Johns Hopkins University Press | isbn=978-1-4214-0352-6 | year=2011 | zbl=1245.00040 }}

* {{ Citation | last1 = Connes | first1 = Alain | author-link = Alain Connes | last2 = Consani | first2 = Caterina | last3 = Marcolli | first3 = Matilde | author3-link=Matilde Marcolli | title = Fun with <math>\mathbb{F}_1</math> | journal = Journal of Number Theory | volume = 129 | issue = 6 | year = 2009 | arxiv = 0806.2401 | pages=1532–1561 |zbl=1228.11143 | mr=2521492 | doi = 10.1016/j.jnt.2008.08.007 }}

* {{Citation | last1 = Connes | first1 = Alain | author-link = Alain Connes | last2 = Consani | first2 = Caterina | title = Schemes over '''F'''₁ and zeta functions | journal = Compositio Mathematica | volume = 146 | number = 6 |publisher = London Mathematical Society | year = 2010 | pages = 1383-14151383–1415 | doi = 10.1112/S0010437X09004692 | arxiv = 0903.2024 }}

* {{ Citation | last1 = Deitmar | first1 = Anton | chapter = Schemes over '''F'''₁ | title = Number Fields and Function Fields: Two Parallel Worlds | editor-last = van der Geer | editor-first = G. | editor2-last = Moonen | editor2-first = B. | editor3-last = Schoof | editor3-first = R. | series = Progress in Mathematics | volume = 239 | year = 2005 }}

* {{ Citation | last1 = Deitmar | first1 = Anton | title = '''F'''₁-schemes and toric varieties | year = 2006 | arxiv = math/0608179 }}

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* {{ Citation | last1 = Tits | first1 = Jacques | author1-link = Jacques Tits|chapter = Sur les analogues algébriques des groupes semi-simples complexes | title = Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain | publisher = Librairie Gauthier-Villars | place = Paris | year = 1957 | pages = 261–289 }}

* {{ Citation | last1 = Toën | first1 = Bertrand | last2 = Vaquié | first2 = Michel | author1-link = Bertrand Toën | title = Au dessous de Spec '''Z''' | arxiv = math/0509684 | year = 2005 }}

* {{ Citation |last1=Vezzani |first1=Alberto |title=Deitmar's versus Toën-Vaquié's schemes over '''F'''₁ |journal=Mathematische Zeitschrift |date=2010 |volume=271 |pages=1-161–16 |doi=10.1007/s00209-011-0896-5 |url=https://www.researchgate.net/publication/225695202_Deitmar's_versus_Toen-Vaquie's_schemes_over_F_1|arxiv=1005.0287 }}

==External links==