# Category: absolute

It is perhaps surprising that Alain Connes and Katia Consani, two icons of noncommutative geometry, restrict themselves to define commutative algebraic geometry over $\mathbb{F}_1$, the field with one element.

My guess of why they stop there is as good as anyone’s. Perhaps they felt that there is already enough noncommutativity in Soule’s gadget-approach (the algebra $\mathcal{A}_X$ as in this post may very well be noncommutative). Perhaps they were only interested in the Bost-Connes system which can be entirely encoded in their commutative $\mathbb{F}_1$-geometry. Perhaps they felt unsure as to what the noncommutative scheme of an affine noncommutative algebra might be. Perhaps …

Remains the fact that their approach screams for a noncommutative extension. Their basic object is a covariant functor

$N~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A)$

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to $N$. In a series of posts on the Connes-Consani paper (starting here) I took some care of getting rid of all scheme-lingo and rephrasing everything entirely into algebras. But then, this set-up can be extended verbatim to noncommuative $\mathbb{F}_1$-geometry, which should start from a covariant functor

$N~:~\mathbf{groups} \rightarrow \mathbf{sets}$

from all finite groups to sets. Let’s recall quickly what the additional info should be making this functor a noncommutative (affine) F_un scheme :

There should be a finitely generated $\mathbb{C}$-algebra $R$ together with a natural transformation (the ‘evaluation’)

$e~:~N \rightarrow \mathbf{maxi}(R) \qquad N(G) \mapsto Hom_{\mathbb{C}-alg}(R, \mathbb{C} G)$

(both $R$ and the group-algebra $\mathbb{C} G$ may be noncommutative). The pair $(N, \mathbf{maxi}(R))$ is then called a gadget and there is an obvious notion of ‘morphism’ between gadgets.

The crucial extra ingredient is an affine $\mathbb{Z}$-algebra (possibly noncommutative) $S$
such that $N$ is a subfunctor of $\mathbf{mini}(S)~:~G \mapsto Hom_{\mathbb{Z}-alg}(S,\mathbb{Z} G)$ together with the following universal property :

any affine $\mathbb{Z}$-algebra $T$ having a gadget-morphism $~(N,\mathbf{maxi}(R)) \rightarrow (\mathbf{mini}(T),\mathbf{maxi}(T \otimes_{\mathbb{Z}} \mathbb{C}))$ comes from a $\mathbb{Z}$-algebra morphism $T \rightarrow S$. (If this sounds too cryptic for you, please read the series on C-C mentioned before).

So, there is no problem in defining noncommutative affine F_un-schemes. However, as with any generalization, this only makes sense provided (a) we get something new and (b) we have interesting examples, not covered by the restricted theory.

At first sight we do not get something new as in the only example we did in the C-C-series (the forgetful functor) it is easy to prove (using the same proof as given in this post) that the forgetful-functor $\mathbf{groups} \rightarrow \mathbf{sets}$ still has as its integral form the integral torus $\mathbb{Z}[x,x^{-1}]$. However, both theories quickly diverge beyond this example.

For example, consider the functor

$\mathbf{groups} \rightarrow \mathbf{sets} \qquad G \mapsto G \times G$

Then, if we restrict to abelian finite groups $\mathbf{abelian}$ it is easy to see (again by a similar argument) that the two-dimensional integer torus $\mathbb{Z}[x,y,x^{-1},y^{-1}]$ is the correct integral form. However, this algebra cannot be the correct form for the functor on the category of all finite groups as any $\mathbb{Z}$-algebra map $\phi~:~\mathbb{Z}[x,y,x^{-1},y^{-1}] \rightarrow \mathbb{Z} G$ determines (and is determined by) a pair of commuting units in $\mathbb{Z} G$, so the above functor can not be a subfunctor if we allow non-Abelian groups.

But then, perhaps there isn’t a minimal integral $\mathbb{Z}$-form for this functor? Well, yes there is. Take the free group in two letters (that is, all words in noncommuting $x,y,x^{-1}$ and $y^{-1}$ satisfying only the trivial cancellation laws between a letter and its inverse), then the corresponding integral group-algebra $\mathbb{Z} \mathcal{F}_2$ does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map $f~:~T \mapsto \mathbb{C} \mathcal{F}_2$ and we have to show, using the universal property that the image of $T$ is contained in the integral group-algebra $\mathbb{Z} \mathcal{F}_2$. Take a generator
$z$ of $T$ then the degree of the image $f(z)$ is bounded say by $d$ and we can always find a subgroup $H \subset \mathcal{F}_2$ such that $\mathcal{F}_2/H$ is a fnite group and the quotient map $\mathbb{C} \mathcal{F}_2 \rightarrow \mathbb{C} \mathcal{F}_2/H$ is injective on the subspace spanned by all words of degree strictly less than $d+1$. Then, the usual diagram-chase finishes the proof.

What makes this work is that the free group $\mathcal{F}_2$ has ‘enough’ subgroups of finite index, a property it shares with many interesting discrete groups. Whence the blurb-message :

if the integers $\mathbb{Z}$ see a discrete group $\Gamma$, then the field $\mathbb{F}_1$ sees its profinite completion $\hat{\Gamma} = \underset{\leftarrow}{lim}~\Gamma/ H$ So, yes, we get something new by extending the Connes-Consani approach to the noncommutative world, but do we have interesting examples? As “interesting” is a subjective qualification, we’d better invoke the authority-argument.

Alexander Grothendieck (sitting on the right, manifestly not disputing a vacant chair with Jean-Pierre Serre, drinking on the left (a marvelous picture taken by F. Hirzebruch in 1958)) was pushing the idea that profinite completions of arithmetical groups were useful in the study of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, via his theory of dessins d’enfants (children;s drawings).

In a previous life, I’ve written a series of posts on dessins d’enfants, so I’ll restrict here to the basics. A smooth projective $\overline{\mathbb{Q}}$-curve $X$ has a Belyi-map $X \rightarrow \mathbb{P}^1_{\overline{\mathbb{Q}}}$ ramified only in three points ${ 0,1,\infty }$. The “drawing” corresponding to $X$ is a bipartite graph, drawn on the Riemann surface $X_{\mathbb{C}}$ obtained by lifting the unit interval $[0,1]$ to $X$. As the absolute Galois group acts on all such curves (and hence on their corresponding drawings), the action of it on these dessins d’enfants may give us a way into the multiple mysteries of the absolute Galois group.

In his “Esquisse d’un programme” (Sketch of a program if you prefer to read it in English) he writes :

“C’est ainsi que mon attention s’est portée vers ce que j’ai appelé depuis la “géométrie algêbrique anabélienne”, dont le point de départ est justement une étude (pour le moment limitée à la caractéristique zéro) de l’action de groupe de Galois “absolus” (notamment les groupes $Gal(\overline{K}/K)$, ou $K$ est une extension de type fini du corps premier) sur des groupes fondamentaux géométriques (profinis) de variétés algébriques (définies sur $K$), et plus particulièrement (rompant avec une tradition bien enracinée) des groupes fondamentaux qui sont trés éloignés des groupes abéliens (et que pour cette raison je nomme “anabéliens”). Parmi ces groupes, et trés proche du groupe $\hat{\pi}_{0,3}$, il y a le compactifié profini du groupe modulaire $SL_2(\mathbb{Z})$, dont le quotient par le centre $\pm 1$ contient le précédent comme sous-groupe de congruence mod 2, et peut s’interpréter d’ailleurs comme groupe “cartographique” orienté, savoir celui qui classifie les cartes orientées triangulées (i.e. celles dont les faces des triangles ou des monogones).”

and a bit further, he writes :

“L’élément de structure de $SL_2(\mathbb{Z})$ qui me fascine avant tout, est bien sur l’action extérieure du groupe de Galois $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ sur le compactifié profini. Par le théorème de Bielyi, prenant les compactifiés profinis de sous-groupes d’indice fini de $SL_2(\mathbb{Z})$, et l’action extérieure induite (quitte à passer également à un sous-groupe overt de $Gal(\overline{\mathbb{Q}},\mathbb{Q})$), on trouve essentiellement les groupes fondamentaux de toutes les courbes algébriques définis sur des corps de nombres $K$, et l’action extérieure de $Gal(\overline{K}/K)$ dessus.”

So, is there a noncommutative affine variety over $\mathbb{F}_1$ of which the unique minimal integral model is the integral group algebra of the modular group $\mathbb{Z} \Gamma$ (with $\Gamma = PSL_2(\mathbb{Z})$? Yes, here it is

$N_{\Gamma}~:~\mathbf{groups} \rightarrow \mathbf{sets} \qquad G \mapsto G_2 \times G_3$

where $G_n$ is the set of all elements of order $n$ in $G$. The reason behind this is that the modular group is the free group product $C_2 \ast C_3$.

Fine, you may say, but all this is just algebra. Where is the noncommutative complex variety or the noncommutative integral scheme in all this? Well, we can introduce them too but as this post is already 1300 words long, I’ll better leave this for another time. In case you cannot stop thinking about it, here’s the short answer.

The complex noncommutative variety has as its ‘points’ all finite dimensional simple complex representations of the modular group, and the ‘points’ of the noncommutative $\mathbb{F}_1$-scheme are exactly the (modular) dessins d’enfants…

In case you haven’t noticed it yet : I’m not living here anymore.

My blogging is (at least for the moment) transfered to the F_un Mathematics blog which some prefer to call the “ceci n’est pas un corps”-blog, which is very fine with me.

Javier gave a talk at MPI on Soule’s approach to algebraic geometry over the elusive field with one element $\mathbb{F}_1$ and wrote two posts about it The skeleton of Soule’s F_un geometry and Gadgets a la Soule. The rough idea being that a variety over the field with one element only acquires flesh after a base extension to $\mathbb{Z}$ and to cyclotomic integers.

I did some posts on a related (but conceptually somewhat easier) approach due to Alain Connes and Katia Consani. I’ve tried to explain their construction at the level of (mature) undergraduate students. So far, there are three posts part1, part2 and part3. Probably there is one more session to come in which I will explain why they need functors to graded sets.

In the weeks to come we plan to post about applications of this F_un-geometry to noncommutative geometry (the Bost-Connes system) and Grothendieck’s anabelian geometry (the theory of dessins d’enfant). I’ll try to leave a short account of the main posts here, but clearly you are invited to feed your feedreader this.

Perhaps I’ll return here for a week mid november to do some old-fashioned vacation blogging. I have to admit I did underestimate Numeo.fr. Rumours have it that our place is connected wirelessly to the web…

At the Max-Planck Institute in Bonn Yuri Manin gave a talk about the field of one element, $\mathbb{F}_1$ earlier this week entitled “Algebraic and analytic geometry over the field F_1”.

Moreover, Javier Lopez-Pena and Bram Mesland will organize a weekly “F_un Study Seminar” starting next tuesday.

Over at Noncommutative Geometry there is an Update on the field with one element pointing us to a YouTube-clip featuring Alain Connes explaining his paper with Katia Consani and Matilde Marcolli entitled “Fun with F_un”. Here’s the clip

Finally, as I’ll be running a seminar here too on F_un, we’ve set up a group blog with the people from MPI (clearly, if you are interested to join us, just tell!). At the moment there are just a few of my old F_un posts and a library of F_un papers, but hopefully a lot will be added soon. So, have a look at F_un mathematics A couple of weeks ago, Alain Connes and Katia Consani arXived their paper “On the notion of geometry over $\mathbb{F}_1$”. Their subtle definition is phrased entirely in Grothendieck‘s scheme-theoretic language of representable functors and may be somewhat hard to get through if you only had a few years of mathematics. I’ll try to give the essence of their definition of an affine scheme over $\mathbb{F}_1$ (and illustrate it with an example) in a couple of posts. All you need to know is what a finite Abelian group is (if you know what a cyclic group is that’ll be enough) and what a commutative algebra is. If you already know what a functor and a natural transformation is, that would be great, but we’ll deal with all that abstract nonsense when we’ll need it.

So take two finite Abelian groups A and B, then a group-morphism is just a map $f~:~A \rightarrow B$ preserving the group-data. That is, f sends the unit element of A to that of B and
f sends a product of two elements in A to the product of their images in B. For example, if $A=C_n$ is a cyclic group of order n with generator g and $B=C_m$ is a cyclic group of order m with generator h, then every groupmorphism from A to B is entirely determined by the image of g let’s say that this image is $h^i$. But, as $g^n=1$ and the conditions on a group-morphism we must have that $h^{in} = (h^i)^n = 1$ and therefore m must divide i.n. This gives you all possible group-morphisms from A to B.

They are plenty of finite abelian groups and many group-morphisms between any pair of them and all this stuff we put into one giant sack and label it $\mathbf{abelian}$. There is another, even bigger sack, which is even simpler to describe. It is labeled $\mathbf{sets}$ and contains all sets as well as all maps between two sets.

Right! Now what might be a map $F~:~\mathbf{abelian} \rightarrow \mathbf{sets}$ between these two sacks? Well, F should map any abelian group A to a set F(A) and any group-morphism $f~:~A \rightarrow B$ to a map between the corresponding sets $F(f)~:~F(A) \rightarrow F(B)$ and do all of this nicely. That is, F should send compositions of group-morphisms to compositions of the corresponding maps, and so on. If you take a pen and a piece of paper, you’re bound to come up with the exact definition of a functor (that’s what F is called).

You want an example? Well, lets take F to be the map sending an Abelian group A to its set of elements (also called A) and which sends a groupmorphism $A \rightarrow B$ to the same map from A to B. All F does is ‘forget’ the extra group-conditions on the sets and maps. For this reason F is called the forgetful functor. We will denote this particular functor by $\underline{\mathbb{G}}_m$, merely to show off.

Luckily, there are lots of other and more interesting examples of such functors. Our first class we will call maxi-functors and they are defined using a finitely generated $\mathbb{C}$-algebra R. That is, R can be written as the quotient of a polynomial algebra

$R = \frac{\mathbb{C}[x_1,\ldots,x_d]}{(f_1,\ldots,f_e)}$

by setting all the polynomials $f_i$ to be zero. For example, take R to be the ring of Laurant polynomials

$R = \mathbb{C}[x,x^{-1}] = \frac{\mathbb{C}[x,y]}{(xy-1)}$

Other, and easier, examples of $\mathbb{C}$-algebras is the group-algebra $\mathbb{C} A$ of a finite Abelian group A. This group-algebra is a finite dimensional vectorspace with basis $e_a$, one for each element $a \in A$ with multiplication rule induced by the relations $e_a.e_b = e_{a.b}$ where on the left-hand side the multiplication . is in the group-algebra whereas on the right hand side the multiplication in the index is that of the group A. By choosing a different basis one can show that the group-algebra is really just the direct sum of copies of $\mathbb{C}$ with component-wise addition and multiplication

$\mathbb{C} A = \mathbb{C} \oplus \ldots \oplus \mathbb{C}$

with as many copies as there are elements in the group A. For example, for the cyclic group $C_n$ we have

$\mathbb{C} C_n = \frac{\mathbb{C}[x]}{(x^n-1)} = \frac{\mathbb{C}[x]}{(x-1)} \oplus \frac{\mathbb{C}[x]}{(x-\zeta)} \oplus \frac{\mathbb{C}[x]}{(x-\zeta^2)} \oplus \ldots \oplus \frac{\mathbb{C}[x]}{(x-\zeta^{n-1})} = \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \oplus \ldots \oplus \mathbb{C}$

The maxi-functor asociated to a $\mathbb{C}$-algebra R is the functor

$\mathbf{maxi}(R)~:~\mathbf{abelian} \rightarrow \mathbf{sets}$

which assigns to a finite Abelian group A the set of all algebra-morphism $R \rightarrow \mathbb{C} A$ from R to the group-algebra of A. But wait, you say (i hope), we also needed a functor to do something on groupmorphisms $f~:~A \rightarrow B$. Exactly, so to f we have an algebra-morphism $f’~:~\mathbb{C} A \rightarrow \mathbb{C}B$ so the functor on morphisms is defined via composition

$\mathbf{maxi}(R)(f)~:~\mathbf{maxi}(R)(A) \rightarrow \mathbf{maxi}(R)(B) \qquad \phi~:~R \rightarrow \mathbb{C} A \mapsto f’ \circ \phi~:~R \rightarrow \mathbb{C} A \rightarrow \mathbb{C} B$

So, what is the maxi-functor $\mathbf{maxi}(\mathbb{C}[x,x^{-1}]$? Well, any $\mathbb{C}$-algebra morphism $\mathbb{C}[x,x^{-1}] \rightarrow \mathbb{C} A$ is fully determined by the image of $x$ which must be a unit in $\mathbb{C} A = \mathbb{C} \oplus \ldots \oplus \mathbb{C}$. That is, all components of the image of $x$ must be non-zero complex numbers, that is

$\mathbf{maxi}(\mathbb{C}[x,x^{-1}])(A) = \mathbb{C}^* \oplus \ldots \oplus \mathbb{C}^*$

where there are as many components as there are elements in A. Thus, the sets $\mathbf{maxi}(R)(A)$ are typically huge which is the reason for the maxi-terminology.

Next, let us turn to mini-functors. They are defined similarly but this time using finitely generated $\mathbb{Z}$-algebras such as $S=\mathbb{Z}[x,x^{-1}]$ and the integral group-rings $\mathbb{Z} A$ for finite Abelian groups A. The structure of these inegral group-rings is a lot more delicate than in the complex case. Let’s consider them for the smallest cyclic groups (the ‘isos’ below are only approximations!)

$\mathbb{Z} C_2 = \frac{\mathbb{Z}[x]}{(x^2-1)} = \frac{\mathbb{Z}[x]}{(x-1)} \oplus \frac{\mathbb{Z}[x]}{(x+1)} = \mathbb{Z} \oplus \mathbb{Z}$

$\mathbb{Z} C_3 = \frac{\mathbb{Z}[x]}{(x^3-1)} = \frac{\mathbb{Z}[x]}{(x-1)} \oplus \frac{\mathbb{Z}[x]}{(x^2+x+1)} = \mathbb{Z} \oplus \mathbb{Z}[\rho]$

$\mathbb{Z} C_4 = \frac{\mathbb{Z}[x]}{(x^4-1)} = \frac{\mathbb{Z}[x]}{(x-1)} \oplus \frac{\mathbb{Z}[x]}{(x+1)} \oplus \frac{\mathbb{Z}[x]}{(x^2+1)} = \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}[i]$

For a $\mathbb{Z}$-algebra S we can define its mini-functor to be the functor

$\mathbf{mini}(S)~:~\mathbf{abelian} \rightarrow \mathbf{sets}$

which assigns to an Abelian group A the set of all $\mathbb{Z}$-algebra morphisms $S \rightarrow \mathbb{Z} A$. For example, for the algebra $\mathbb{Z}[x,x^{-1}]$ we have that

$\mathbf{mini}(\mathbb{Z} [x,x^{-1}]~(A) = (\mathbb{Z} A)^*$

the set of all invertible elements in the integral group-algebra. To study these sets one has to study the units of cyclotomic integers. From the above decompositions it is easy to verify that for the first few cyclic groups, the corresponding sets are $\pm C_2, \pm C_3$ and $\pm C_4$. However, in general this set doesn’t have to be finite. It is a well-known result that the group of units of an integral group-ring of a finite Abelian group is of the form

$(\mathbb{Z} A)^* = \pm A \times \mathbb{Z}^{\oplus r}$

where $r = \frac{1}{2}(o(A) + 1 + n_2 -2c)$ where $o(A)$ is the number of elements of A, $n_2$ is the number of elements of order 2 and c is the number of cyclic subgroups of A. So, these sets can still be infinite but at least they are a lot more manageable, explaining the mini-terminology.

Now, we would love to go one step deeper and define nano-functors by the same procedure, this time using finitely generated algebras over $\mathbb{F}_1$, the field with one element. But as we do not really know what we might mean by this, we simply define a nano-functor to be a subfunctor of a mini-functor, that is, a nano-functor N has an associated mini-functor $\mathbf{mini}(S)$ such that for all finite Abelian groups A we have that $N(A) \subset \mathbf{mini}(S)(A)$.

For example, the forgetful functor at the beginning, which we pompously denoted $\underline{\mathbb{G}}_m$ is a nano-functor as it is a subfunctor of the mini-functor $\mathbf{mini}(\mathbb{Z}[x,x^{-1}])$.

Now we are allmost done : an affine $\mathbb{F}_1$-scheme in the sense of Connes and Consani is a pair consisting of a nano-functor N and a maxi-functor $\mathbf{maxi}(R)$ such that two rather strong conditions are satisfied :

• there is an evaluation ‘map’ of functors $e~:~N \rightarrow \mathbf{maxi}(R)$
• this pair determines uniquely a ‘minimal’ mini-functor $\mathbf{mini}(S)$ of which N is a subfunctor

of course we still have to turn this into proper definitions but that will have to await another post. For now, suffice it to say that the pair $~(\underline{\mathbb{G}}_m,\mathbf{maxi}(\mathbb{C}[x,x^{-1}]))$ is a $\mathbb{F}_1$-scheme with corresponding uniquely determined mini-functor $\mathbf{mini}(\mathbb{Z}[x,x^{-1}])$, called the multiplicative group scheme.

Continued here Amidst all LHC-noise, Yuri I. Manin arXived today an interesting paper Cyclotomy and analytic geometry over $\mathbb{F}_1$.

The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element $\mathbb{F}_1$ (in French called ‘F-un’). I have tried to do a couple of posts on F-un some time ago but now realize, reading Manin’s paper, I may have given up way too soon…

At several places in the paper, Manin hints at a possible noncommutative geometry over $\mathbb{F}_1$ :

This is the appropriate place to stress that in a wider context of Toen-Vaqui ‘Au-dessous de Spec Z’, or eventually in noncommutative $\mathbb{F}_1$-geometry, teh spectrum of $\mathbb{F}_1$ loses its privileged position as a final object of a geometric category. For example, in noncommutative geometry, or in an appropriate category of stacks, the quotient of this spectrum modulo the trivial action of a group must lie below this spectrum.

Soule’s algebras $\mathcal{A}_X$ are a very important element of the structure, in particular, because they form a bridge to Arakelov geometry. Soule uses concrete choices of them in order to produce ‘just right’ supply of morphisms, without a priori constraining these choices formally. In this work, we use these algebras and their version also to pave a way to the analytic (and possibly non-commutative) geometry over $\mathbb{F}_1$.

Back when I was writing the first batch of F-un posts, I briefly contemplated the possibility of a noncommutative geometry over $\mathbb{F}_1$, but quickly forgot about it because I thought it would be forced to reduce to commutative geometry.

Here is the quick argument : noncommutative geometry is really the study of coalgebras (see for example my paper or if you prefer more trustworthy sources the Kontsevich-Soibelman paper). Now, unless I made a mistake, I think all coalgebras over $\mathbb{F}_1$ must be co-commutative (even group-like), so reducing to commutative geometry.

Surely, I’m missing something…