nLab Borger's absolute geometry




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 RR, hence on its spectrum Spec(R)Spec()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(𝔽 1)Spec(\mathbb{F}_1) (which is defined thereby). This definition yields an essential geometric morphism of gros etale toposes

Et(Spec())Et(Spec(𝔽 1)), 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 toposEt(Spec(𝔽 1))Et(Spec(\mathbb{F}_1)) which is rich and genuinely interesting in its own right.


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]k[z] over a finite field kk, as well as to the ring 𝒪 \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]k[z] and 𝒪 \mathcal{O}_{\mathbb{C}}, witnessing their affine-ness in one variable, is that they carry a canonical derivation, namely z\frac{\partial}{\partial z}, and that the ground field is recovered as the quotient by the ideal generated by zz. More to the point, for each maximal ideal (zx)(z-x) there is the first order translation operator id+(zx)(zx)\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 xx.

Therefore if \mathbb{Z} is analogous to k[z]k[z] and to 𝒪 \mathcal{O}_{\mathbb{C}}, then it ought to admit analogous operators, one for each of its maximal ideals (p)(p) given by a prime number pp \in \mathbb{Z}. Remarkably, such a collection of operations indeed exists on \mathbb{Z}: the pp-power operations () p:(-)^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:nn p=n+p pn. (-)^p : n \mapsto n^p = n + p \cdot\partial_p n \,.

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

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

Hence by analogy it makes sense to think of p:\partial_p : \mathbb{Z} \to \mathbb{Z} as being like a derivation on Spec()\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]k[z]-algebra AA is to be thought of as a space Spec(A)\mathrm{Spec}(A) over Spec(k[z])\mathrm{Spec}(k[z]). The canonical derivation on the latter canonically lifts to the former, and is given by the same formula: id+(zx)(zx)\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}. This exhibits the fact that Spec(A)Spec(k[z])\mathrm{Spec}(A) \to \mathrm{Spec}(k[z]) is simply a product of Spec(A/(z))Spec(k)\mathrm{Spec}(A/(z)) \to \mathrm{Spec}(k) with the affine line 𝔸 k=Spec(k[z])\mathbb{A}_k = \mathrm{Spec}(k[z]) over the ground field kk. Analogously, any commutative ring RR is to be thought of as a space Spec(R)\mathrm{Spec}(R) over Spec()\mathrm{Spec}(\mathbb{Z}) and the above arithmetic translation operator canonically lifts to Spec(R)\mathrm{Spec}(R), by the same formula: aa pa \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 RR is an 𝔽 p\mathbb{F}_p-algebra, so that Spec(R)\mathrm{Spec}(R) is supported over the point Spec(𝔽 p)\mathrm{Spec}(\mathbb{F}_p) in Spec()\mathrm{Spec}(\mathbb{Z}), then the standard fact that in characteristic pp the pp-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 Frob p\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.

Spec(R) Frob p+0 Spec(R) Spec(𝔽 p) id+0 Spec(𝔽 p) Spec() id+p p Spec() \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=id+p p(-)^p = \mathrm{id} + p \cdot \partial_p on Spec()\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 Spec(R)\mathrm{Spec}(R) is to be a choice of ring homomorphism Φ:RR\Phi : R \to R (a “finite translation”) in addition to a derivation-like operation p:RR\partial_p : R \to R (the “infinitesimal translation”), such that

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

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

p Φ() pΦp:RR. \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.

Spec(R) Φ+p p Φ Spec(R) Spec() id+p p Spec() \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 RR equipped with such an endomorphism Φ\Phi is equivalently a Lambda ring (an insight highlighted by James Borger). In this way a Λ\Lambda-ring structure on a commutative ring RR is the arithmetic geometry-analog of exhibiting Spec(R)Spec()\mathrm{Spec}(R) \to \mathrm{Spec}(\mathbb{Z}) as being like a product by the affine line over the non-existent field 𝔽 1\mathbb{F}_1 of a map to “Spec(𝔽 1)\mathrm{Spec}(\mathbb{F}_1)”. This should be one way to think of how Λ\Lambda-rings embody “geometry over 𝔽 1\mathbb{F}_1” as proposed in (Borger 09).

Finally notice that the refinement of the () p(-)^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)


under construction

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

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

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


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

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

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

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

is the gros etale topos of arithmetic geometry.

Put a compatible Grothendieck topology on ΛRing op\Lambda Ring^{op}(…) and write the resulting site Spec(𝔽 1) etSpec(\mathbb{F}_1)_{et}.

In analogy we write

Et(Spec(𝔽 1))Sh(Spec(𝔽 1) et) Et(Spec(\mathbb{F}_1)) \coloneqq Sh(Spec(\mathbb{F}_1)_{et})

and speak of the “etale topos over Spec(𝔽 1)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() et)Symm *Symm *U *Symm !U *Witt *U !Witt *Witt !PSh(Spec(𝔽 1) et) 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() et)Symm !U *Witt *U !Witt *Witt !Sh(Spec(𝔽 1) et) 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 *W *)(U !Witt !U !Witt *). (W^\ast \dashv W_\ast) \coloneqq ( U_! \circ Witt_! \dashv U_! \circ Witt_\ast) \,.

infinitesimal disk bundle\dashv jet comonad


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

This is based on technical details laid out in

More discussion relating to this includes

Related discussion of arithmetic jet spaces is in

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

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

Last revised on February 9, 2020 at 17:27:13. See the history of this page for a list of all contributions to it.