Connes-Consani for undergraduates (3)

A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element , which by Grothendieck’s scheme-philosophy should determine a functor

from finite Abelian groups to sets, typically giving pretty small sets . Using the F_un mantra that should be an algebra over any -variety determines an integral scheme by extension of scalars, as well as a complex variety (by extending further to ). We have already connected the complex variety with the original functor into a gadget that is a couple where is the coordinate ring of a complex affine variety having the property that every element of can be realized as a -point of . Ringtheoretically this simply means that to every element there is an algebra map .

Today we will determine which gadgets determine an integral scheme, and do so uniquely, and call them the sought for affine schemes over .

Let’s begin with our example : being the forgetful functor, that is for every finite Abelian group, then the complex algebra partners up to form a gadget because to every element there is a natural algebra map defined by sending . Clearly, there is an obvious integral form of this complex algebra, namely but we have already seen that this algebra represents the mini-functor

and that the group of units of the integral group ring usually is a lot bigger than . So, perhaps there is another less obvious -algebra doing a much better job at approximating ? That is, if we can formulate this more precisely…

In general, every -algebra defines a gadget with the obvious (that is, extension of scalars) evaluation map

Right, so how might one express the fact that the integral affine scheme with integral algebra is the ‘best’ integral approximation of a gadget . Well, to begin its representing functor should at least contain the information given by , that is, is a sub-functor of (meaning that for every finite Abelian group we have a natural inclusion ). As to the “best”-part, we must express that all other candidates factor through . That is, suppose we have an integral algebra and a morphism of gadgets (as defined last time)

then there ought to be -algebra morphism such that the above map factors through an induced gadget-map .

Fine, but is this definition good enough in our trivial example? In other words, is the “obvious” integral ring the best integral choice for approximating the forgetful functor ? Well, take any finitely generated integral algebra , then saying that there is a morphism of gadgets from to means that there is a -algebra map such that for every finite Abelian group we have a commuting diagram

Here, is the natural evaluation map defined before sending a group-element to the algebra map defined by and the vertical map on the right-hand side is extensions by scalars. From this data we must be able to show that the image of the algebra map

is contained in the integral subalgebra . So, take any generator of then its image is a Laurent polynomial of degree say (that is, with all coefficients a priori in and we need to talk them into ).

Now comes the basic trick : take a cyclic group of order , then the above commuting diagram applied to the generator of (the evaluation of which is the natural projection map ) gives us the commuting diagram

where the horizontal map is the natural inclusion map. Tracing along the diagram we see that indeed all coefficients of have to be integers! Applying the same argument to the other generators of (possibly for varying values of N) we see that , indeed, and hence that is the best integral approximation for .

That is, we have our first example of an affine variety over the field with one element : .

What makes this example work is that the infinite group (of which the complex group-algebra is the algebra ) has enough finite Abelian group-quotients. In other words, doesn’t see but rather its profinite completion … (to be continued when we’ll consider noncommutative -schemes)

In general, an affine -scheme is a gadget with morphism of gadgets provided that the integral algebra is the best integral approximation in the sense made explicit before. This rounds up our first attempt to understand the Connes-Consani approach to define geometry over apart from one important omission : we have only considered functors to , whereas it is crucial in the Connes-Consani paper to consider more generally functors to graded sets. In the final part of this series we’ll explain what that’s all about.