nLab Borger's absolute geometry

Idea

One proposal for a precise realization of the idea of “absolute” arithmetic geometry over Spec(F1) is Borger’s absolute geometry (Borger 09). Here the structure of a Lambda-ring on a ring $R$, hence on its spectrum $Spec(R) \to Spec(\mathbb{Z})$, 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 ring of Witt vectors construction and may be thought of as producing arithmetic jet spaces. In this sense the adjoint triple here would be directly analogous to the base change along the unit of an infinitesimal shape modality whose induced comonad is the jet comonad.

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.

Motivation

The following is an attempt to motivate or make intuitively clear why lifts of Frobenius morphisms may be related to “absolute geometry” over F1.

First of all, the function field analogy says that $\mathbb{Z}$ is analogous to the polynomial ring $k[z]$ over a finite field $k$, as well as to the ring $\mathcal{O}_{\mathbb{C}}$ of (entire) holomorphic functions on the complex plane.

To make this analogy more concrete, notice that one characteristic property of the rings $k[z]$ and $\mathcal{O}_{\mathbb{C}}$, witnessing their affine-ness in one variable, is that they carry a canonical derivation, namely $\frac{\partial}{\partial z}$, and that the ground field is recovered as the quotient by the ideal generated by $z$. More to the point, for each maximal ideal $(z-x)$ there is the first order translation operator $\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}$ and the quotient by its difference from the identity is the residue field of functions at the point $x$.

Therefore if $\mathbb{Z}$ is analogous to $k[z]$ and to $\mathcal{O}_{\mathbb{C}}$, then it ought to admit analogous operators, one for each of its maximal ideals $(p)$ given by a prime number $p \in \mathbb{Z}$. Remarkably, such a collection of operations indeed exists on $\mathbb{Z}$: the $p$-power operations $(-)^p : \mathbb{Z} \to \mathbb{Z}$ (acting on the underlying set of $\mathbb{Z}$) which by Fermat's little theorem is indeed of the above form

$$(-)^p : n \mapsto n^p = n + p \cdot\partial_p n \,.$$

Here the expression $\partial_p n \in \mathbb{Z}$ is uniquely defined by this equation, it is given by the Fermat quotient operation

$$\partial_p : n \mapsto \partial_p n \coloneqq \frac{n^p - n}{p} \,.$$

Hence by analogy it makes sense to think of $\partial_p : \mathbb{Z} \to \mathbb{Z}$ as being like a derivation on $\mathrm{Spec}(\mathbb{Z})$ – it is called a p-derivation. This is the beginning of the theory of arithmetic differential equations (Buium 05).

More generally, a $k[z]$-algebra $A$ is to be thought of as a space $\mathrm{Spec}(A)$ over $\mathrm{Spec}(k[z])$. The canonical derivation on the latter canonically lifts to the former, and is given by the same formula: $\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}$. This exhibits the fact that $\mathrm{Spec}(A) \to \mathrm{Spec}(k[z])$ is simply a product of $\mathrm{Spec}(A/(z)) \to \mathrm{Spec}(k)$ with the affine line $\mathbb{A}_k = \mathrm{Spec}(k[z])$ over the ground field $k$. Analogously, any commutative ring $R$ is to be thought of as a space $\mathrm{Spec}(R)$ over $\mathrm{Spec}(\mathbb{Z})$ and the above arithmetic translation operator canonically lifts to $\mathrm{Spec}(R)$, by the same formula: $a \mapsto a^p$. However, inspecting this one finds that not only is the derivation-like part lifted non-trivially, but also the identity-part is lifted in general to some homomorphism: in particular if $R$ is an $\mathbb{F}_p$-algebra, so that $\mathrm{Spec}(R)$ is supported over the point $\mathrm{Spec}(\mathbb{F}_p)$ in $\mathrm{Spec}(\mathbb{Z})$, then the standard fact that in characteristic $p$ the $p$-power operation is a ring homomorphism now means that the derivation-like action vanishes here, as expected from the analogy, but that a pure homomorphism part covering the identity remains – the Frobenius homomorphism $\mathrm{Frob}_p$. If by some abuse of notation we allow ourselves to write formally dual morphisms for maps between rings that are not necessarily homomorphisms, then this situation is, possibly, usefully visualized as follows.

$$\array{ \mathrm{Spec}(R) &\stackrel{\mathrm{Frob}_p + 0}{\longrightarrow} & \mathrm{Spec}(R) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{F}_p) &\stackrel{\mathrm{id} + 0}{\longrightarrow} & \mathrm{Spec}(\mathbb{F}_p) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{Z}) &\stackrel{\mathrm{id} + p \cdot \partial_p}{\longrightarrow} & \mathrm{Spec}(\mathbb{Z}) }$$

This suggests that we are to think of the operation $(-)^p = \mathrm{id} + p \cdot \partial_p$ on $\mathrm{Spec}(\mathbb{Z})$ not just as analogous to the identity transformation plus a derivation, but as analogous to the sum of a general finite transformation plus a derivation. In other words, a lift of this operation to some $\mathrm{Spec}(R)$ is to be a choice of ring homomorphism $\Phi : R \to R$ (a “finite translation”) in addition to a derivation-like operation $\partial_p : R \to R$ (the “infinitesimal translation”), such that

$$(-)^p = \Phi + p \cdot\partial_p \,.$$

As before, when $p$ is invertible in $R$, hence away from the fiber over $\mathrm{Spec}(\mathbb{F}_p)$, then $\partial_p$ is uniquely fixed by this equation once $\Phi$ is chosen, and hence it is $\Phi$ alone which is to be chosen on $R$. A pair $(R,\Phi)$ satisfying the above equation is equivalently a “$\Lambda_p$-ring” (see at Lambda ring). Or rather, by the analogy we are to lift the whole collection of operators $\mathrm{id} + p \cdot\partial_p$ for all primes $p$ and hence are to ask that $\Phi$ satisfies the above equation for all $p$ and hence defines for all primes $p$ the derivation-like operator (function on the underlying set of $R$)

$$\partial^\Phi_p \coloneqq \frac{(-)^p - \Phi}{p} : R \to R \,.$$

This is the general form of an “arithmetic differential operator”, see for instance the first pages of (Buium 13) for review.

In terms of the abuse of notation already employed previously, this situation is usefully visualized as follows.

$$\array{ \mathrm{Spec}(R) &\stackrel{\Phi + p\cdot \partial^{\Phi}_p}{\longrightarrow} & \mathrm{Spec}(R) \\ \downarrow && \downarrow \\ \mathrm{Spec}(\mathbb{Z}) &\stackrel{\mathrm{id} + p\cdot \partial_p}{\longrightarrow} & \mathrm{Spec}(\mathbb{Z}) }$$

A ring $R$ equipped with such an endomorphism $\Phi$ is equivalently a Lambda ring (an insight highlighted by James Borger). In this way a $\Lambda$-ring structure on a commutative ring $R$ is the arithmetic geometry-analog of exhibiting $\mathrm{Spec}(R) \to \mathrm{Spec}(\mathbb{Z})$ as being like a product by the affine line over the non-existent field $\mathbb{F}_1$ of a map to “$\mathrm{Spec}(\mathbb{F}_1)$”. This should be one way to think of how $\Lambda$-rings embody “geometry over $\mathbb{F}_1$” as proposed in (Borger 09).

Finally notice that the refinement of the $(-)^p$-operation from arithmetic geometry to E-infinity arithmetic geometry is given by the power operations in multiplicative cohomology theory (Lurie, remark 2.2.9)

Definition

under construction

Write CRing for the category of (finitely generated) commutative rings and $\Lambda Ring$ for that of Lambda-rings.

By the discussion at here the forgetful functor $U \;\colon\; \Lambda Ring \longrightarrow CRing$ from Lambda-rings to commutative rings has

• a left adjoint, given by forming the ring $Symm$ of symmetric functions;

• a right adjoint given by forming the ring of Witt vectors $W$.

$$(Symm \dashv U \dashv Witt) \;\colon\; \Lambda Ring \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\overset{Witt}{\leftarrow}}} CRing \,.$$

Hence

• rings of Witt vectors are the co-free Lambda-rings;

• rings of symmetric functions are the free Lambda-rings.

Regarding $Ring^{op}$ as a site for arithmetic geometry, the order of the adjoints is reversed by forming opposite categories

$$(Witt \dashv U \dashv Symm) \;\colon\; CRing^{op} \stackrel{\overset{Witt}{\longrightarrow}}{ \stackrel{\overset{U}{\longleftarrow}}{ \overset{Symm}{\longrightarrow} } } \Lambda Ring^{op} \,.$$

Write $Spec(\mathbb{Z})_{et}$ for $CRing^{op}$ equipped with the etale topology. Hence

$$Et(Spec(\mathbb{Z})) \coloneqq Sh(Spec(\mathbb{Z})_{et})$$

is the gros etale topos of arithmetic geometry.

Put a compatible Grothendieck topology on $\Lambda Ring^{op}$(…) and write the resulting site $Spec(\mathbb{F}_1)_{et}$.

In analogy we write

$$Et(Spec(\mathbb{F}_1)) \coloneqq Sh(Spec(\mathbb{F}_1)_{et})$$

and speak of the “etale topos over $Spec(\mathbb{F}_1)$”, or the “absolute base topos” or something like this.

The above adjoint triple on sites then induces a sequence of adjoint functors on the categories of presheaves by left and right Kan extension

$$PSh(Spec(\mathbb{Z})_{et}) \stackrel{\stackrel{Witt_!}{\longrightarrow}}{ \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{ \stackrel{\stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow}}{ \stackrel{\stackrel{Symm^\ast \simeq U_\ast}{\longleftarrow}}{ \stackrel{Symm_\ast}{\longrightarrow} } } } } PSh(Spec(\mathbb{F}_1)_{et})$$

The top three restrict to sheaves

$$Sh(Spec(\mathbb{Z})_{et}) \stackrel{\stackrel{Witt_!}{\longrightarrow}}{ \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{ \stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow} } } Sh(Spec(\mathbb{F}_1)_{et})$$

The induced adjoint pair of monad/comonad is

$$(W^\ast \dashv W_\ast) \coloneqq ( U_! \circ Witt_! \dashv U_! \circ Witt_\ast) \,.$$

infinitesimal disk bundle$\dashv$ jet comonad

References

The original article which proposes the topos over Lambda-rings as a realization of F1-geometry is

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

This is based on technical details laid out 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)

Related discussion of arithmetic jet spaces is in

• Alexandru Buium, Arithmetic differential equations, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005.

• Alexandru Buium, Differential calculus with integers (arXiv:1308.5194, slighly differing pdf)

Related discussion of power operations in E-infinity arithmetic geometry is around remark 2.2.9 of

• Jacob Lurie, Rational and p-adic Homotopy Theory

Further discussion of and speculation on an analogy between power operations and Borger’s absolute geometry is in

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

• Jack Morava, Rakha Santhanam, Power operations and Absolute geometry (pdf)