Connes-Consani for undergraduates (3)

Posted by on Oct 10, 2008 in Connes-Consani2008, undergraduate1 comment

A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element \mathbb{F}_1, which by Grothendieck’s scheme-philosophy should determine a functor

\wis{nano}(N)~:~\wis{abelian} \rightarrow \wis{sets} \qquad A \mapsto N(A)

from finite Abelian groups to sets, typically giving pretty small sets N(A). Using the F_un mantra that \Z should be an algebra over \mathbb{F}_1 any \mathbb{F}_1-variety determines an integral scheme by extension of scalars, as well as a complex variety (by extending further to \C). We have already connected the complex variety with the original functor into a gadget that is a couple ~(\wis{nano}(N),\wis{maxi}(R)) where R is the coordinate ring of a complex affine variety X_R having the property that every element of N(A) can be realized as a \C A-point of X_R. Ringtheoretically this simply means that to every element x \in N(A) there is an algebra map N_x~:~R \rightarrow \C A.

Today we will determine which gadgets determine an integral scheme, and do so uniquely, and call them the sought for affine schemes over \mathbb{F}_1.

Let’s begin with our example : \wis{nano}(N) = \underline{\mathbb{G}}_m being the forgetful functor, that is N(A)=A for every finite Abelian group, then the complex algebra R= \C[x,x^{-1}] partners up to form a gadget because to every element a \in N(A)=A there is a natural algebra map N_a~:~\C[x,x^{-1}] \rightarrow \C A defined by sending x \mapsto e_a. Clearly, there is an obvious integral form of this complex algebra, namely \Z[x,x^{-1}] but we have already seen that this algebra represents the mini-functor

\wis{min}(\Z[x,x^{-1}])~:~\wis{abelian} \rightarrow \wis{sets} \qquad A \mapsto (\Z A)^*

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

In general, every \Z-algebra S defines a gadget \wis{gadget}(S) = (\wis{mini}(S),\wis{maxi}(S \otimes_{\Z} \C)) with the obvious (that is, extension of scalars) evaluation map

\wis{mini}(S)(A) = Hom_{\Z-alg}(S, \Z A) \rightarrow Hom_{\C-alg}(S \otimes_{\Z} \C, \C A) = \wis{maxi}(S \otimes_{\Z} \C)(A)

Right, so how might one express the fact that the integral affine scheme X_T with integral algebra T is the ‘best’ integral approximation of a gadget ~(\wis{nano}(N),\wis{maxi}(R)). Well, to begin its representing functor should at least contain the information given by N, that is, \wis{nano}(N) is a sub-functor of \wis{mini}(T) (meaning that for every finite Abelian group A we have a natural inclusion N(A) \subset Hom_{\Z-alg}(T, \Z A)). As to the “best”-part, we must express that all other candidates factor through T. That is, suppose we have an integral algebra S and a morphism of gadgets (as defined last time)

f~:~(\wis{nano}(N),\wis{maxi}(R)) \rightarrow \wis{gadget}(S) = (\wis{mini}(S),\wis{maxi}(S \otimes_{\Z} \C))

then there ought to be \Z-algebra morphism T \rightarrow S such that the above map f factors through an induced gadget-map \wis{gadget}(T) \rightarrow \wis{gadget}(S).

Fine, but is this definition good enough in our trivial example? In other words, is the “obvious” integral ring \Z[x,x^{-1}] the best integral choice for approximating the forgetful functor N=\underline{\mathbb{G}}_m? Well, take any finitely generated integral algebra S, then saying that there is a morphism of gadgets from ~(\underline{\mathbb{G}}_m,\wis{maxi}(\C[x,x^{-1}]) to \wis{gadget}(S) means that there is a \C-algebra map \psi~:~S \otimes_{\Z} \C \rightarrow \C[x,x^{-1}] such that for every finite Abelian group A we have a commuting diagram

\xymatrix{A \ar[rr] \ar[d]_e & & Hom_{\Z-alg}(S, \Z A) \ar[d] \\
Hom_{\C-alg}(\C[x,x^{-1}],\C A) \ar[rr]^{- \circ \psi} & & Hom_{\C-alg}(S \otimes_{\Z} \C, \C A)}

Here, e is the natural evaluation map defined before sending a group-element a \in A to the algebra map defined by x \mapsto e_a 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

\xymatrix{S \ar[r]^{i} & S \otimes_{\Z} \C \ar[r]^{\psi} & \C[x,x^{-1}]}

is contained in the integral subalgebra \Z[x,x^{-1}]. So, take any generator z of S then its image \psi(z) \in \C[x,x^{-1}] is a Laurent polynomial of degree say d (that is, \psi(z) = c_{-d} x^{-d} + \hdots c_{-1} x^{-1} + c_0 + c_1 x + \hdots + c_d x^d with all coefficients a priori in \C and we need to talk them into \Z).

Now comes the basic trick : take a cyclic group A=C_N of order N > d, then the above commuting diagram applied to the generator of C_N (the evaluation of which is the natural projection map \pi~:~\C[x.x^{-1}] \rightarrow \C[x,x^{-1}]/(x^N-1) = \C C_N) gives us the commuting diagram

\xymatrix{S \ar[r] \ar[d] & S \otimes_{\Z} \C \ar[r]^{\psi} & \C[x,x^{-1}] \ar[d]^{\pi} \\
\Z C_n = \frac{\Z[x,x^{-1}]}{(x^N-1)} \ar[rr]^j & & \frac{\C[x,x^{-1}]}{(x^N-1)}}

where the horizontal map j is the natural inclusion map. Tracing z \in S along the diagram we see that indeed all coefficients of \psi(z) have to be integers! Applying the same argument to the other generators of S (possibly for varying values of N) we see that , indeed, \psi(S) \subset \Z[x,x^{-1}] and hence that \Z[x,x^{-1}] is the best integral approximation for \underline{\mathbb{G}}_m.

That is, we have our first example of an affine variety over the field with one element \mathbb{F}_1 : ~(\underline{\mathbb{G}}_m,\wis{maxi}(\C[x,x^{-1}]) \rightarrow \wis{gadget}(\Z[x,x^{-1}]).

What makes this example work is that the infinite group \Z (of which the complex group-algebra is the algebra \C[x,x^{-1}]) has enough finite Abelian group-quotients. In other words, \mathbb{F}_1 doesn’t see \Z but rather its profinite completion \hat{\Z} = \underset{\leftarrow} \Z/N\Z… (to be continued when we’ll consider noncommutative \mathbb{F}_1-schemes)

In general, an affine \mathbb{F}_1-scheme is a gadget with morphism of gadgets ~(\wis{nano}(N),\wis{maxi}(R)) \rightarrow \wis{gadget}(S) provided that the integral algebra S 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 \mathbb{F}_1 apart from one important omission : we have only considered functors to \wis{sets}, 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.

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