nLab field with one element

Overview

Various phenomena in the context of algebraic geometry/arithmetic geometry (and particularly in the context of algebraic groups) over finite fields $\mathbb{F}_q$ turn out to make perfect sense as expressions in $q$ when extrapolated to the case $q=1$, and to reflect interesting (combinatorial, representation theoretical…) facts, even though, of course, there is no actual field with a single element (since in a field by definition the elements 1 and 0 are distinct).

Motivated by such observations, Jacques Tits envisioned in (Tits 57) a new kind of geometry adapted to the explanation of these identities. Christophe Soulé then expanded on Tits’ ideas by introducing the notion of field with one element and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of field, the idea is that there is some other object, denoted $\mathbb{F}_1$, such that it does make sense to speak of “geometry over $\mathbb{F}_1$”. Following the French pronunciation one also writes $F_{un}$ (and is thus led to the inevitable pun).

In the relative point of view the $S$-schemes are schemes with a morphism of schemes over a base scheme $S$; but every $S$-scheme is a scheme over Spec(Z). In absolute algebraic geometry all “generalized schemes” should live over $Spec(F_1)$ and $Spec(F_1)$ should live below $Spec(\mathbb{Z})$; this is similar to the fact that the quotient stacks like $[*/G]$ live below the single point $*$ (there is a direct image functor from sheaves on a point to sheaves over $[*/G]$). One of the principal and very bold hopes is that the study of $F_{un}$ should lead to a natural proof of Riemann conjecture (see also MathOverflow here). It was originally suggested by (Manin 95) that the Riemann hypothesis might be solved by finding an $\mathbb{F}_1$-analogue of André Weil‘s proof for the case of arithmetic curves over the finite fields $\mathbb{F}_q$.

A first proposal for what a variety “over $\mathbb{F}_1$” ought to be is due to (Soulé 04). After that a plethora of further proposals appeared, including (Connes-Consani 08).

Borger’s absolute geometry

Maybe an emerging consensus is that the preferred approach is Borger's absolute geometry (Borger 09). Here the structure of a Lambda-ring on a ring $R$, hence on $Spec(R) \to Spec(\mathbb{Z})$, is interpreted as a collection of lifts of all Frobenius morphisms and hence as descent data for descent to $Spec(\mathbb{F}_1)$ (which is defined thereby). This definition yields an essential geometric morphism of gros etale toposes

$$Et(Spec(\mathbb{Z})) \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}} Et(Spec(\mathbb{F}_1)) \,,$$

where on the right the notation is just suggestive, the topos is a suitable one over Lambda-rings. Here the middle inverse image is the forgetful functor which forgets the Lambda structure, and its right adjoint direct image is given by the arithmetic jet space construction (via the ring of Witt vectors construction).

This proposal seems to subsume many aspects of other existing proposals (see e.g. Le Bruyn 13) and stands out as yielding an “absolute base topos” $Et(Spec(\mathbb{F}_1))$ which is rich and genuinely interesting in its own right.

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Algebra over $\mathbb{F}_1$

Modules/Vector spaces over $\mathbb{F}_1$

One proposal here is to identify the concept of finite rank modules/finite-dimensional vector spaces over the field with one element with that of (pointed) finite sets

$$\mathbb{F}_1 Mod_{fin} \;\simeq\; Set^{\ast/}_{fin}$$

and hence the symmetric group $\Sigma_n$ on $n$ elements with the general linear group over $\mathbb{F}_1$:

$$GL(n,\mathbb{F}_1) \;\simeq\; \Sigma_n$$

(e.g. Cohn 04, “puzzle 1”, Durov 07, 2.5.6, Snyder 07)

But it has also been argued by Connes and Consani (see Beardsley & Nakamura 24, 2.3) that we should understand $\mathbb{F}_1$-modules as Segal $\Gamma$-sets, i.e., pointed functors $Fin_{\ast} \to Set_{\ast}$.

Algebraic K-theory

With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from above it follows that the algebraic K-theory over $\mathbb{F}_1$ is stable cohomotopy:

$$\begin{aligned} K \mathbb{F}_1 & \coloneqq\; K(\mathbb{F}_1 Mod) \\ & \simeq\; K(FinSet) \\ & \simeq \mathbb{S} \end{aligned} \,.$$

Here in the second step we used the definition of algebraic K-theory for ordinary commutative rings as the K-theory of the permutative category of modules (this example), in the second step we used the identification of modules over $\mathbb{F}_1$ with pointed finite sets from above, and finally we used the identification of the K-theory of the permutative category of finite set with the sphere spectrum (this example), which is the spectrum representing stable cohomotopy, by definition.

The perspective that the K-theory $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be stable Cohomotopy has been highlighted in (Deitmar 06, p. 2, Guillot 06, Mahanta 17, Dundas-Goodwillie-McCarthy 13, II 1.2, Morava, Connes-Consani 16).), and fully explicitly in Chu, Lorscheid & Santhanam 10, Thm. 5.9 and Beardsley & Nakamura 2024, Cor. 2.25. (Chu et al. also generalize to equivariant stable Cohomotopy and equivariant K-theory.)

With the understanding of $\mathbb{F}_1$-modules as $\Gamma$-sets, $K \mathbb{F}_1$ remains equal to the sphere spectrum.

(equivariant) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$	Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant) complex cobordism cohomology	MU	$MU_G(\ast)$	completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant) algebraic K-theory	$K \mathbb{F}_p$	representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$	Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

Related concepts

• Lambda-ring,

• blue scheme

• tropical geometry.

References

After the very first observations by Tits, pioneers were Christophe Soulé and Kapranov and Smirnov. More recently there are extensive works by Alain Connes and Katia Consani, Nikolai Durov, James Borger and Oliver Lorscheid.

Expositions

• Henry Cohn, Projective geometry over $\mathbb{F}_1$ and the Gaussian binomial coefficients, American Mathematical Monthly 111 (2004), 487-495 (arXiv:math/0407093)

• Lieven Le Bruyn, Looking for $F_{un}$, blog

• Noah Snyder, The field with one element, 2007

• Javier López Peña, Oliver Lorscheid, Mapping $F_1$-land:An overview of geometries over the field with one element, arXiv/0909.0069

• John Baez, This Week’s Finds 259 (html blog)

• Alain Connes, Fun with $\mathbf{F}_1$, 5 min. video

• Lieven Le Bruyn, The field with one element, seminar notes 2008 (web)

• Oliver Lorscheid, Lectures on $\mathbb{F}_1$, 2014 (pdf)

• Oliver Lorscheid, $\mathbb{F}_1$ for everyone, 2018 (arXiv:1801.05337)

• Tim Hosgood, Under $\mathop{Spec}\mathbb{Z}$: a reader’s companion, master thesis, 2016 (pdf).

Original articles

• Jacques Tits, Sur les analogues algebriques des groupes semi-simples complexes. In Colloque d’algebre superieure, tenu a Bruxelles du 19 au 22 decembre 1956, Centre Belge de Recherches Mathematiques, pages 261-289. Etablissements Ceuterick, Louvain, 1957.

• Christophe Soulé, Les varietes sur le corps a un element Mosc. Math. J., 4(1):217-244, 312, 2004 (pdf)

• Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa) Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar (pdf)

• Bertrand Toen, Michel Vaquie, Under Spec Z (arXiv:math/0509684)

Around (0.4.24.2) in

• Nikolai Durov, New Approach to Arakelov Geometry (arXiv:0704.2030)

the algebraic structure of $\mathbb{F}_1$ is regarded as being the maybe monad, hence modules over $\mathbb{F}_1$ are defined to be monad-algebras over the maybe monad, hence pointed sets.

Other approaches include

• Alain Connes, Caterina Consani, Matilde Marcolli, Fun with $\mathbf{F}_1$, arxiv/0806.2401

• Yuri Manin, Cyclotomy and analytic geometry over $F_1$, arxiv/0809.1564

• Alain Connes, Caterina Consani, On the notion of geometry over $\F_1$, arxiv/0809.2926;

  Schemes over $\F_1$ and zeta functions, arxiv/0903.2024; Characteristic one, entropy and the absolute point, in: Noncommutative Geometry, Arithmetic, and Related Topics, 21st Meeting of the Japan-U.S. Math. Inst., Baltimore 2009, JHUP (2012), pp. 75–139, arxiv/0911.3537;

  From monoids to hyperstructures: in search of an absolute arithmetic, arxiv/1006.4810;

  On the arithmetic of the BC-system, arxiv/1103.4672; Projective geometry in characteristic one and the epicyclic category, arxiv/1309.0406

• M. Marcolli, Ryan Thorngren, Thermodynamical semirings, arXiv/1108.2874

• Bora Yalkinoglu, On Endomotives, Lambda-rings and Bost-Connes systems, With an appendix by Sergey Neshveyev, arxiv/1105.5022

• Jonathan Beardsley, So Nakamura, Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules [arXiv:2404.04730]

The approach in terms of Lambda-rings due to

• James Borger, Lambda-rings and the field with one element (arXiv/0906.3146)

with details in

• James Borger, The basic geometry of Witt vectors, I: The affine case (arXiv:0801.1691)

• James Borger, The basic geometry of Witt vectors, II: Spaces (arXiv:1006.0092)

More discussion relating to this includes

• Lieven Le Bruyn, Absolute geometry and the Habiro topology (arXiv:1304.6532)

Relation to stable homotopy theory

The interpretation of stable cohomotopy as algebraic K-theory over $\mathbb{F}_1$ is amplified in the following articles:

• Anton Deitmar, Remarks on zeta functions and K-theory over $\mathbb{F}_1$ (arXiv:math/0605429)

• Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)

• Jack Morava, Rekha Santhanam, Power operations and absolute geometry, 2012 (pdf)

• Snigdhayan Mahanta, G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 (arXiv:1110.6001)

• John D. Berman, p. 92 of Categorified algebra and equivariant homotopy theory (arXiv:1805.08745)

• Chenghao Chu, Oliver Lorscheid, Rekha Santhanam, Sheaves and K-theory for $\mathbb{F}_1$-schemes, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 (arxiv:1010.2896)

see also

• Jack Morava, Some background on Manin’s theorem $K(\mathbb{F}_1) \sim \mathbb{S}$ (pdf, MoravaSomeBackground.pdf)

• Alain Connes, Caterina Consani, Absolute algebra and Segal’s Gamma sets, Journal of Number Theory 162 (2016): 518-551 (arXiv:1502.05585)

In quantum field theory

Speculation on quantum field theory over $\mathbb{F}_1$ (via wonderful compactifications of configuration spaces of points)

• Dori Bejleri, Matilde Marcolli, Quantum field theory over $\mathbb{F}_1$, Journal of Geometry and Physics 69 (2013) 40-59 [arXiv:1209.4837, doi:10.1016/j.geomphys.2013.03.002]

• Seyed Khaki, Original $\mathbb{F}_1$ in emergent spacetime [arXiv:2401.07822]