# noncommutative F_un geometry (1)

Posted by on Oct 13, 2008 in Connes-Consani2008, research4 comments

It is perhaps surprising that Alain Connes and Katia Consani, two icons of noncommutative geometry, restrict themselves to define commutative algebraic geometry over , 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 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 -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

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to . 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 -geometry, which should start from a covariant functor

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 -algebra together with a natural transformation (the ‘evaluation’)

(both and the group-algebra may be noncommutative). The pair is then called a gadget and there is an obvious notion of ‘morphism’ between gadgets.

The crucial extra ingredient is an affine -algebra (possibly noncommutative) such that is a subfunctor of together with the following universal property :

any affine -algebra having a gadget-morphism comes from a -algebra morphism . (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 still has as its integral form the integral torus . However, both theories quickly diverge beyond this example.

For example, consider the functor

Then, if we restrict to abelian finite groups it is easy to see (again by a similar argument) that the two-dimensional integer torus 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 -algebra map determines (and is determined by) a pair of commuting units in , so the above functor can not be a subfunctor if we allow non-Abelian groups.

But then, perhaps there isn’t a minimal integral -form for this functor? Well, yes there is. Take the free group in two letters (that is, all words in noncommuting and satisfying only the trivial cancellation laws between a letter and its inverse), then the corresponding integral group-algebra does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map and we have to show, using the universal property that the image of is contained in the integral group-algebra . Take a generator of then the degree of the image is bounded say by and we can always find a subgroup such that is a fnite group and the quotient map is injective on the subspace spanned by all words of degree strictly less than . Then, the usual diagram-chase finishes the proof.

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

if the integers see a discrete group , then the field sees its profinite completion

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 , 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 -curve has a Belyi-map ramified only in three points . The “drawing” corresponding to is a bipartite graph, drawn on the Riemann surface obtained by lifting the unit interval to . 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 :

and a bit further, he writes :

So, is there a noncommutative affine variety over of which the unique minimal integral model is the integral group algebra of the modular group (with ? Yes, here it is

where is the set of all elements of order in . The reason behind this is that the modular group is the free group product .

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 -scheme are exactly the (modular) dessins d’enfants…

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