The Field with One Element

Introduction: Analogies between Function Fields and Number Fields

As has been hinted at, and mentioned in passing, in several previous posts on this blog, there are important analogies between numbers and functions. The analogy can perhaps be made most explicit in the case of $\mathbb{Z}$ (the ring of ordinary integers) and $\mathbb{F}_{p}[t]$ (the ring of polynomials in one variable $t$ over the finite field $\mathbb{F}_{p}$). We also often say that the analogy is between $\mathbb{Q}$ (the field of rational numbers) and $\mathbb{F}_{p}(t)$ (the field of rational functions in one variable $t$ over the finite field $\mathbb{F}_{p}$), which are the respective fields of fractions of $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$. Recall also from Some Basics of Class Field Theory that $\mathbb{Q}$ and $\mathbb{F}_{p}(t)$ are examples of what we call global fields, together with their respective finite extensions.

Let us go back to $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$ and compare their similarities. They are both principal ideal domains, which means that all their ideals can be generated by a single element. They both have groups of units (elements which have multiplicative inverses) which are finite. They both have an infinite number of prime ideals (generated by prime numbers in the case of $\mathbb{Z}$, and by monic irreducible polynomials in the case of $\mathbb{F}_{p}[t]$), and finally, they share the property that their residue fields over these prime ideals are finite.

But of course, despite all these analogies, a rather obvious question still remains unanswered. Regarding this question we quote the words of the mathematician Christophe Soule:

“The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that $\text{Spec}(\mathbb{Z})$ (with a point at infinity added, as is familiar in Arakelov geometry) is like a (complete) curve, but over which field?”

This question led to the development of the idea of the “field with one element”, also written $\mathbb{F}_{1}$, or sometimes $\mathbb{F}_{\text{un}}$ (it’s a pun taken from “un”, the French word for “one”). Taken literally, there is no such thing  as a “field” with one element – the way we define a field, it must always have a “one” and a “zero”, and these two elements must be different. Instead, the idea of the “field with one element” is just a name for ideas that extend the analogy between function fields and number fields, as if this “field” really existed. The name itself has historical origins in the work of the mathematician Jacques Tits involving certain groups called Chevalley groups and Weil groups, where surprising results appear in the limit when the number of elements of the finite fields involved goes to one – but in most approaches now, the “field with one element” is not a field, and often has more than one element. The whole point is that these ideas may still work, even though the “field” itself may not even exist! As one might expect, in order to pursue these ideas one must think out of the box, and different mathematicians have approached this question in different ways.

In this post, we will look at four approaches to the field with one element, developed by the mathematicians Anton Deitmar, Christophe Soule, Bertrand Toen and Michel Vaquie, and James Borger. There are many more approaches besides these, but we will perhaps discuss them in future posts.

Note: Throughout this post it will be helpful to remind ourselves that since there exists a map from the integers $\mathbb{Z}$ to any ring, we can think of rings as $\mathbb{Z}$-algebras. One of the ways the idea of the field with one element is approached is by exploring what $\mathbb{F}_{1}$-algebras mean, if ordinary rings are $\mathbb{Z}$-algebras.

The Approach of Deitmar

Deitmar defines the “category of rings over $\mathbb{F}_{1}$” (this is the term Deitmar uses, but we can also think of this as the category of $\mathbb{F}_{1}$-algebras) as simply the category of monoids. A monoid $A$ is also written as $\mathbb{F}_{A}$ to emphasize its nature as a “ring over $\mathbb{F}_{1}$“. The “field with one element” $\mathbb{F}_{1}$ is simply defined to be the trivial monoid.

For an $\mathbb{F}_{1}$-ring $\mathbb{F}_{A}$ we define the base extension (see Grothendieck’s Relative Point of View) to $\mathbb{Z}$ by taking the “monoid ring” $\mathbb{Z}[A]$:

$\displaystyle \mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[A]$

We may think of this monoid ring as a ring whose elements are formal sums of elements of the monoid $A$ with integer coefficients, and with a multiplication provided by the multiplication on $A$, commuting with the scalar multiplication.

Meanwhile we also have the forgetful functor $F$ which simply “forgets” the additive structure of a ring, leaving us with a monoid under its multiplication operation. The base extension functor $-\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ is left adjoint to the forgetful functor $F$, i.e. for every ring $R$ and every $\mathbb{F}_{A}/\mathbb{F}_{1}$ we have $\text{Hom}_{\text{Rings}}(\mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}, R)\cong\text{Hom}_{\mathbb{F}_{1}}(\mathbb{F}_{A},F(R))$ (see also Adjoint Functors and Monads).

The important concepts of “localization” and “ideals” in the theory of rings, important to construct the structure sheaf of a variety or a scheme, have analogues in the theory of monoids. The idea is that they only make use of the multiplicative structure of rings, so we can forget the additive structure and consider monoids instead. Hence, we can define varieties or schemes over $\mathbb{F}_{1}$. Many other constructions of algebraic geometry can be replicated with only monoids instead of rings, such as sheaves of modules over the structure sheaf. Deitmar then defines the zeta function of a scheme over $\mathbb{F}_{1}$, and hopes to connect this with known ideas about zeta functions (see for example our discussion in The Riemann Hypothesis for Curves over Finite Fields).

Deitmar’s idea of using monoids is one of the earlier approaches to the idea of the field with one element, and has become somewhat of a template for other approaches. One may be able to notice the influence of Deitmar’s work in the other approaches that we will discuss in this post.

The Approach of Soule

Soule’s question, as phrased in his paper On the Field with One Element, is as follows:

“Which varieties over $\mathbb{Z}$ are obtained by base change from $\mathbb{F}_{1}$ to $\mathbb{Z}$?”

Soule’s approach to answering this question then makes use of three concepts. The first one is a suggestion from the early days of the development of the idea of the field with one element, apparently due to the mathematicians Andre Weil and Kenkichi Iwasawa, that the finite field extensions of the field with one element should consist of the roots of unity, together with zero.

The second concept is an important point that we only touched on briefly from Algebraic Spaces and Stacks, namely, that we may identify the functor of points of a scheme with the scheme itself. Now the functor of points of a scheme is uniquely determined by its values on affine schemes, and the category of affine schemes is the opposite category to the category of rings; therefore, we now redefine a scheme simply as a covariant functor from the category of rings to the category of sets, which is representable.

The third concept is the idea of an evaluation of a function at a point. Soule implements this concept by including a $\mathbb{C}$-algebra as part of his definition of a variety over $\mathbb{F}_{1}$, together with a natural transformation that expresses this evaluation.

We now give the details of Soule’s construction, proceeding in four steps. Taking into account the first concept mentioned earlier,  we consider the following expression, the base extension of $\mathbb{F}_{1^{n}}$ to $\mathbb{Z}$ over $\mathbb{F}_{1}$:

$\displaystyle \mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[T]/(T^{n}-1)=\mathbb{Z}[\mu_{n}]$

We shall also denote this ring by $R_{n}$. We can form a category whose objects are the finite tensor products of $R_{n}$, for $n\geq 1$, and we denote this category by $\mathcal{R}$.

An affine gadget over $\mathbb{F}_{1}$ is a triple $(\underline{X},\mathcal{A}_{X},e_{X})$ where $\underline{X}$ is a covariant functor from the category $\mathcal{R}$ to the category of sets, $\mathcal{A}_{X}$ is a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{X}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of affine gadgets consists of a natural transformation $\underline{\phi}:\underline{X}\rightarrow\underline{Y}$ and a morphism of algebras $\phi^{*}:\mathcal{A}_{X}\rightarrow\mathcal{A}_{Y}$ such that $f(\underline{\phi}(P))=(\phi^{*}(f))(P)$. A morphism $(\underline{\phi}, \phi^{*})$ is also called an immersion if $\underline{\phi}$ and $\phi^{*}$ are both injective.

An affine variety over $\mathbb{F}_{1}$ is an affine gadget $X=(\underline{X},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{X}(R)$ is finite, and

(ii) there exists an affine scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{G}(X_{\mathbb{Z}})$ with the universal property that for any other affine scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{G}(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{G}(V)$ such that $\varphi=\mathcal{G}(\varphi_{\mathbb{Z}})\circ i$.

An object over $\mathbb{F}_{1}$ is a triple $(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ where $\underline{\underline{X}}$ is a contravariant functor from the category of affine gadgets over $\mathbb{F}_{1}$$\mathcal{A}_{X}$ is once again a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{\underline{X}}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of objects is defined in the same way as a morphism of affine gadgets.

A variety over $\mathbb{F}_{1}$ is an object $X=(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{\underline{X}}(\text{Spec}(R))$ is finite, and

(ii) there exists a scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{O}b(X_{\mathbb{Z}})$ with the universal property that for any other scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{O}b(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{O}b(V)$ such that $\varphi=\mathcal{O}b(\varphi_{\mathbb{Z}})\circ i$.

Like Deitmar, Soule constructs the zeta function of a variety over $\mathbb{F}_{1}$, and furthermore explores connections with certain kinds of varieties called “toric varieties”, which are also of interest in other approaches to the field with one element, and the theory of motives (see The Theory of Motives).

The Approach of Toen and Vaquie

We recall from Grothendieck’s Relative Point of View that we call a scheme $X$ a scheme “over” $S$, or an $S$-scheme, if there is a morphism of schemes from $X$ to $S$, and if $S$ is an affine scheme defined as $\text{Spec}(R)$ for some ring $R$, we also refer to it as a scheme over $R$, or an $R$-scheme. We recall also every scheme is a scheme over $\text{Spec}(\mathbb{Z})$, or a $\mathbb{Z}$-scheme. The approach of Toen and Vaquie is to construct categories of schemes “under” $\text{Spec}(\mathbb{Z})$.

From Monoidal Categories and Monoids we know that rings are the monoid objects in the monoidal category of abelian groups, and abelian groups are $\mathbb{Z}$-modules.

More generally, for a symmetric monoidal category $(\textbf{C}, \otimes, \mathbf{1})$ that is complete, cocomplete, and closed (i.e. possesses internal Homs related to the monoidal structure $\otimes$, see again Monoidal Categories and Monoids), we have in $\textbf{C}$ a notion of monoid, for such a monoid $A$ a notion of an $A$-module, and for a morphism of monoids $A\rightarrow B$ a notion of a base change functor $-\otimes_{A}B$ from $A$-modules to $B$-modules.

Therefore, if we have a category $\textbf{C}$ with a symmetric monoidal functor $\textbf{C}\rightarrow \mathbb{Z}\text{-Mod}$, we obtain a notion of a “scheme relative to $\textbf{C}$” and a base change functor to $\mathbb{Z}$-schemes. This gives us our sought-for notion of schemes under $\text{Spec}(\mathbb{Z})$.

In particular, there exists a notion of commutative monoids (associative and with unit) in $\textbf{C}$, and they form a category which we denote by $\textbf{Comm}(\textbf{C})$. We define the category of affine schemes related to $\textbf{C}$ as $\textbf{Aff}_{\textbf{C}}:= \textbf{Comm}(\textbf{C})^{\text{op}}$.

These constructions satisfy certain properties needed to define a category of schemes relative to $(\textbf{C},\otimes,\mathbf{1})$, such as a notion of Zariski topology. A relative scheme is defined as a sheaf on the site $\textbf{Aff}_{\textbf{C}}$ provided with the Zariski topology, and which has a covering by affine schemes. The category of schemes obtained is denoted $\textbf{Sch}(\textbf{C})$. It is a subcategory of the category of sheaves on $\textbf{Aff}_{\textbf{C}}$ which is closed under the formation of fiber products and disjoint unions. It contains a full subcategory of affine schemes, given by the representable sheaves, and which is equivalent to the category $\textbf{Comm}(\textbf{C})^{\text{op}}$. The purely categorical nature of the construction makes the category $\textbf{Sch}(\textbf{C})$ functorial in $\textbf{C}$.

In their paper, Toen and Vacquie give six examples of their construction, one of which is just the ordinary category of schemes, while the other five are schemes “under $\text{Spec}(\mathbb{Z})$“.

First we let $(C,\otimes,\mathbf{1})=(\mathbb{Z}\text{-Mod},\otimes,\mathbb{Z})$, the symmetric monoidal category of abelian groups (for the tensor product). The category of schemes obtained $\mathbb{Z}\text{-Sch}$ is equivalent to the category of schemes in the usual sense.

The second example will be $(C,\otimes,\mathbf{1})=(\mathbb{N}\text{-Mod},\otimes,\mathbb{N})$ the category of commutative monoids, or abelian semigroups, with the tensor product, which could also be called $\mathbb{N}$-modules. The category of schemes in this case will be denoted $\mathbb{N}\text{-Sch}$, and the subcategory of affine schemes is equivalent to the opposite category of commutative semirings.

The third example is $(C,\otimes,\mathbf{1})=(\text{Ens},\times, *)$, the symmetric monoidal category of sets with the direct product. The category of relative schemes will be denoted $\mathbb{F}_{1}\text{-Sch}$, and we can think of them as schemes or varieties defined on the field with one element. By definition, the subcategory of affine $\mathbb{F}_{1}$-schemes is equivalent to the opposite category of commutative monoids.

We have the base change functors

$-\otimes_{\mathbb{F}_{1}}\mathbb{N}:\mathbb{F}_{1}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{N}}\mathbb{Z}:\mathbb{N}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

We can compose these base change functors and represent it with the following diagram:

$\text{Spec}(\mathbb{Z})\rightarrow\text{Spec}(\mathbb{N})\rightarrow\text{Spec}(\mathbb{F}_{1})$.

The final three examples of “schemes under $\text{Spec}(\mathbb{Z})$” given by Toen and Vaquie make use of ideas from “homotopical algebraic geometry“. Homotopical algebraic geometry is a very interesting subject that unfortunately we have not discussed much on this blog. Roughly, in homotopical algebraic geometry the role of rings in ordinary algebraic geometry is taken over by ring spectra – spectra (in the sense of Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) with a “smash product” operation. This allows us to make use of concepts from abstract homotopy theory. In this post we will only introduce some very basic concepts that we will need to discuss Toen and Vaquie’s examples, and leave the rest to the references.

We will need the concepts of $\Gamma$-spaces and simplicial sets. We define the category $\Gamma^{0}$ to be the category whose objects are “pointed” finite sets (a finite set where one element is defined to be the “basepoint”) and whose morphisms are maps of finite sets that preserve the basepoint. We also define the category $\Delta$ to be the category whose objects are finite ordered sets $[n]=\{0<1<2... and whose morphisms are monotone (non-decreasing) maps of finite ordered sets. A $\Gamma$-space is then simply a covariant functor from the category $\Gamma^{0}$ to the category of pointed sets, while a simplicial set is a covariant functor from the category $\Delta$ to the category of sets. Simplicial sets are rather abstract constructions, but they are inspired by simplices and simplicial complexes in algebraic topology (see Simplices).

Let $M$ be a $\Gamma$-space. If there is a monoid structure on $\pi_{0}M(1_{+})$ (see Homotopy Theory), then we say that $M$ is a special $\Gamma$-space. If, in addition, this structure is also an abelian group structure, then we say that $M$ is a very special $\Gamma$-space.

The category of $\Gamma$-spaces and the category of simplicial sets are both symmetric monoidal categories, which we need to define relative schemes. For the category of $\Gamma$-spaces, we have the smash product, defined by the requirement that any morphism $F_{1}\wedge F_{2}\rightarrow G$ to any functor $G$ from $\Gamma^{0}\times \Gamma^{0}$ to the category of pointed sets be a natural transformation, i.e. there are maps of pointed sets from $F_{1}\wedge F_{2}(X\wedge Y)$ to $G(X\wedge Y)$, natural in $X$ and $Y$ (here $X\wedge Y$ refers to the smash product of pointed sets obtained by taking the direct product and collapsing the wedge sum, see Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories),  and the unit is the sphere spectrum $\mathbb{S}$, which is just the inclusion functor from the category of pointed finite sets to the category of pointed sets.

For the category of simplicial sets, we have the direct product, defined as the functor $X\times Y$ which sends the finite ordered set $[n]$ to the set $X([n])\times Y([n])$, for two simplicial sets $X$ and $Y$, and the unit is the functor $*$, which sends any finite ordered set to the set with a single element.

We now go back to Toen and Vaquie’s final three examples of relative schemes. The first of these examples is when one has $(C,\otimes,\mathbf{1}) = (\mathcal{GS},\wedge,\mathbb{S})$, the category of very special $\Gamma$-spaces. We thus have a category of schemes relative to $\mathcal{GS}$, which we will denote $\mathbb{S}\text{-Sch}$, where the notation $\mathbb{S}$ recalls the sphere spectrum.

The second example is $(C,\otimes,\mathbf{1})=(\mathcal{MS},\wedge,\mathbb{S}_{+})$, the category of special $\Gamma$-spaces. The category of relative schemes will be noted $\mathbb{S}_{+}\text{-Sch}$, and its affine objects are homotopical analogs of commutative semirings. The notation $\mathbb{S}_{+}$ intuitively means the semiring in spectra of positive integers, and is a homotopical version of the semiring $\mathbb{N}$.

The third example is $(C,\otimes,\mathbf{1})=(\text{SEns},\times,*)$, the category of simplicial sets with its direct product. The schemes that we obtain are homotopical versions of the $\mathbb{F}_{1}$-schemes, and will be called $\mathbb{S}_{1}$-schemes, where $\mathbb{S}_{1}$ may be thought of as the “ring spectrum with one element”, in analogy with $\mathbb{F}_{1}$, the “field with one element”.

Similar to the earlier cases, we also have the base change functors

$-\otimes_{\mathbb{S}_{1}}\mathbb{S}_{+}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{S}_{+}\text{-Sch}$

and

$-\otimes_{\mathbb{S}_{+}}\mathbb{S}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{S}\text{-Sch}$

which we can also compose and represent it with the following diagram:

$\text{Spec}(\mathbb{S})\rightarrow\text{Spec}(\mathbb{S}_{+})\rightarrow\text{Spec}(\mathbb{S}_{1})$.

Moreover, we also have the following functors:

$-\otimes_{\mathbb{S}_{1}}\mathbb{F}_{1}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{F}_{1}\text{-Sch}$

$-\otimes_{\mathbb{S}_{+}}\mathbb{N}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{S}}\mathbb{Z}:\mathbb{S}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

which relate the “homotopical” relative schemes to the ordinary relative schemes. And so, all these schemes, both the new schemes “under $\text{Spec}(\mathbb{Z})$” as well as the ordinary schemes over $\text{Spec}(\mathbb{Z})$, are related to each other.

The Approach of Borger

Borger’s approach makes use of the idea of adjoint triples (see Adjoint Functors and Monads). Before we discuss the field with one element in this approach, let us first discuss something more elementary. Consider a field $K$ and and a field extension $L$ of $K$, and let $G=\text{Gal}(L/K)$. We have the following adjoint triple:

$\displaystyle \text{Weil restrict}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

$\displaystyle -\otimes_{K}L: K\textbf{-Alg}\rightarrow L\textbf{-Alg}$

$\displaystyle \text{forget base}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

Grothendieck’s abstract reformulation of Galois theory says that there is an equivalence of categories between the category of $K$-algebras and the category of $L$-algebras with an action of $G$. This means that we can also consider the above adjoint triple in the following sense:

$\displaystyle A\rightarrow\otimes_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

$\displaystyle \text{fgt}: L\textbf{-Alg}\text{ (with }G\text{-action)}\rightarrow L\textbf{-Alg}$

$\displaystyle A\rightarrow\prod_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

Let us now go back to the field with one element. We want to construct the following adjoint triple:

$\displaystyle \text{Weil restrict}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

$\displaystyle -\otimes_{\mathbb{F}_{1}}\mathbb{Z}:\mathbb{F}_{1}\textbf{-Alg}\rightarrow\mathbb{Z}\textbf{-Alg}$

$\displaystyle \text{forget base}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

Following the above example of the field $K$ and the field extension $L$ of $K$, we will approach the construction of this adjoint triple by considering instead the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

We must now discuss the meaning of the concepts involved in the last adjoint triple. In particular, the must define the category of $\Lambda$-rings, as well as the adjoint functors $\Lambda\odot-$, $\text{fgt}$, and $W(-)$ that form the adjoint triple.

Let $R$ be a ring and let $p$ be a prime number. A Frobenius lift is a ring homomorphism $\psi_{p}:R\rightarrow R$ such that $F\circ q=q\circ\psi_{p}$ where $q:R\rightarrow R/pR$ is the quotient map and $F:R/pR\rightarrow R/pR$ is the Frobenius map which sends an element $x$ to the element $x^{p}$.

Closely related to the idea of Frobenius lifts is the idea of $p$-derivations. If the terminology is reminiscent of differential calculus, this is because Borger’s approach is closely related to the mathematician Alexandru Buium’s theory of “arithmetic differential equations“. If numbers are like functions, then what Buium wants to figure out is what the analogue of a derivative of a function should be for numbers.

Let

$\displaystyle \psi_{p}(x)=x^{p}+p\delta_{p}(x)$.

Being a ring homomorphism means that $\psi$ satisfies the following properties:

(1) $\psi_{p}(x+y)=\psi_{p}(x)+\psi_{p}(y)$

(2) $\psi_{p}(xy)=\psi_{p}(x)\psi_{p}(y)$

(3) $\psi_{p}(1)=1$

(4) $\psi_{p}(0)=0$

Recalling that $\psi_{p}(x)=x^{p}+p\delta_{p}(x)$, this means that $\delta_{p}(x)$ must satisfy the following properties corresponding to the above properties for $\psi_{p}(x)$:

(1) $\delta_{p}(x+y)=\delta_{p}(x)+\delta_{p}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

(2) $\delta_{p}(xy)=x^{p}\delta_{p}(y)+y^{p}\delta_{p}(x)+p\delta_{p}(x)\delta_{p}(y)$

(3) $\delta_{p}(1)=0$

(4) $\delta_{p}(0)=0$.

Let

$\displaystyle \Lambda_{p}\odot A=\mathbb{Z}[\delta_{p}^{\circ n}(a)|n\geqslant 0,a\in A]/\sim$

where $\sim$ is the equivalence relation given by the “Liebniz rule”, i.e.

$\displaystyle \delta_{p}^{\circ 0}(x+y)=\delta_{p}^{\circ 0}(x)+\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 0}(xy)=\delta_{p}^{\circ 0}(x)\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 1}(x+y)=\delta_{p}^{\circ 1}(x)+\delta_{p}^{\circ 1}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

$\displaystyle \delta_{p}^{\circ 1}(xy)=\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)+p\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)$

and so on.

We now discuss the closely related (and an also important part of modern mathematical research)  notion of Witt vectors. We define the ring of Witt vectors of the ring $A$ by

$\displaystyle W_{p}(A)=A\times A\times...$

with ring operations given by

$\displaystyle (a_{0},a_{1},...)+(b_{0},b_{1},...)=(a_{0}+b_{0},a_{1}+b_{1}-\sum_{i=1}^{p-1}\frac{1}{p}\binom{p}{i}a_{0}^{i}b_{0}^{p-i},...)$

$\displaystyle (a_{0},a_{1},...)(b_{0},b_{1},...)=(a_{0}b_{0},a_{0}^{p}b_{1}+a_{1}b_{0}^{p}+pa_{1}b_{1},...)$

$\displaystyle 0=(0,0,...)$

$\displaystyle 1=(1,0,...)$

The functors

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

A $\Lambda_{p}$-ring is defined to be the smallest $\Lambda_{p}^{'}$-ring that contains $e$, where a $\Lambda_{p}^{'}$-ring is in turn defined to be a $p$-torsion free ring together with a Frobenius lift. But it so happens that a $\Lambda_{p}$-ring is also the same thing as a $\delta_{p}$-ring, so we also have the following adjoint triple:

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda_{p}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

Now that we know the basics of a “$p$-typical” $\Lambda$-ring, which is a ring together with a Frobenius morphism $\psi_{p}$ for one fixed $p$, we can also consider a ring together with a Frobenius morphism $\psi_{p}$ for every prime $p$, to form a “big” $\Lambda$-ring. We will then obtain the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

This is just the adjoint triple that we were looking for in the beginning of this section. In other words, we now have what we need to construct rings over $\mathbb{F}_{1}$, or $\mathbb{F}_{1}$-algebras and moreover, we have an adjoint triple that relates them to ordinary rings (or $\mathbb{Z}$-algebras).

We can then generalize these constructions from rings to schemes. The definition of a $\Lambda$-structure on general schemes is complicated and left to the references, but when the scheme $X$ is flat over $\mathbb{Z}$ (see The Hom and Tensor Functors), a $\Lambda$-structure on $X$ is simply defined to be a commuting family of endomorphisms $\psi_{p}$, one for each prime $p$, such that they agree with the $p$-th power Frobenius map on the fibers $X\times_{\text{Spec}(\mathbb{Z})}\mathbb{F}_{p}$.

One may notice that in Borger’s approach an $\mathbb{F}_{1}$-scheme has more structure than a $\mathbb{Z}$-scheme, whereas in Deitmar’s approach $\mathbb{F}_{1}$-schemes, being commutative monoids, have less structure than $\mathbb{Z}$-schemes. One may actually think of the $\Lambda$-structure as “descent data” to $\mathbb{F}_{1}$. In other words, the $\Lambda$-structure tells us how a scheme defined over $\mathbb{Z}$ is defined over $\mathbb{F}_{1}$. There is actually a way to use a monoid $M$ to construct a $\Lambda$-ring $\mathbb{Z}[M]$, where $\mathbb{Z}[M]$ is just the monoid ring as described earlier in the approach of Deitmar, and the Frobenius lifts are defined by $\psi_{p}=m^{p}$ for $m\in M$. We therefore have some sort of connection between Deitmar’s approach (which is also easily seen to be closely related to Soule’s and Toen and Vaquie’s approach) with Borger’s approach.

Conclusion

We have mentioned only four approaches to the idea of the field with one element in this rather lengthy post. There are many others, and these approaches are often related to each other. In addition, there are other approaches to uncovering even more analogies between function fields and number fields that are not commonly classified as being part of this circle of ideas. To end this post, we just mention that many open problems in mathematics, such as the abc conjecture and the Riemann hypothesis, have function field analogues that have already been solved (we have already discussed the function field analogue of the Riemann hypothesis in The Riemann Hypothesis for Curves over Finite Fields) – perhaps an investigation of these analogies would lead to the solution of their number field analogues – or, in the other direction, perhaps work on these problems would help uncover more aspects of these mysterious and beautiful analogies.

References:

Field with One Element on Wikipedia

Field with One Element on the nLab

Function Field Analogy on the nLab

Schemes over F1 by Anton Deitmar

Lectures on Algebraic Varieties over F1 by Christophe Soule

Les Varieties sur le Corps a un Element by Christophe Soule

On the Field with One Element by Christophe Soule

Under Spec Z by Bertrand Toen and Michel Vaquie

Lambda-Rings and the Field with One Element by James Borger

Witt Vectors, Lambda-Rings, and Arithmetic Jet Spaces by James Borger

Mapping F1-Land: An Overview of Geometries over the Field with One Element by Javier Lopez-Pena and Oliver Lorscheid

Geometry and the Absolute Point by Lieven Le Bruyn

This Week’s Finds in Mathematical Physics (Week 259) by John Baez

Algebraic Number Theory by Jurgen Neukirch

The Local Structure of Algebraic K-Theory by Bjorn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy