Connes-Consani for undergraduates (2)

Tags: Connes, Consani, F_un, geometry, groups

October 6, 2008 0

Last time we have seen how an affine $\mathbb{C}$ -algebra R gives us a maxi-functor (because the associated sets are typically huge)

$\wis{maxi}(R)~:~\wis{abelian} \rightarrow \wis{sets} \qquad A \mapsto Hom_{\C-alg}(R, \C A)$

Substantially smaller sets are produced from finitely generated $\Z$ -algebras S (therefore called mini-functors)

$\wis{mini}(S)~:~\wis{abelian} \rightarrow \wis{sets} \qquad A \mapsto Hom_{\Z-alg}(S, \Z A)$

Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety $X_R = \wis{max}(R)$ (the set of maximal ideals of the algebra R) with complex coordinate ring R and for a mini-functor by the integral affine scheme $X_S = \wis{spec}(S)$ (the set of all prime ideals of the algebra S).

The ‘philosophy’ of Fun mathematics is that an object over this virtual field with one element $\mathbb{F}_1$ records the essence of possibly complicated complex- or integral- objects in a small combinatorial thing.

For example, an n-dimensional complex vectorspace $\C^{n}$ has as its integral form a lattice of rank n $\Z^{\oplus n}$ . The corresponding $\mathbb{F}_1$ -objects only records the dimension n, so it is a finite set consisting of n elements (think of them as the set of base-vectors of the vectorspace).

Similarly, all base-changes of the complex vectorspace $\C^n$ are given by invertible matrices with complex coefficients $GL_n(\C)$ . Of these base-changes, the only ones leaving the integral lattice $\Z^{\oplus n}$ intact are the matrices having all their entries integers and their determinant equal to $\pm 1$ , that is the group $GL_n(\Z)$ . Of these integral matrices, the only ones that shuffle the base-vectors around are the permutation matrices, that is the group of all possible ways to permute the n base-vectors. In fact, this example also illustrates Tits’ original motivation to introduce $\mathbb{F}_1$ : the finite group is the Weyl-group of the complex Lie group $GL_n(\C)$ .

So, we expect a geometric $\mathbb{F}_1$ -object to determine a much smaller functor from finite abelian groups to sets, and, therefore we call it a nano-functor

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

but as we do not know yet what the correct geometric object might be we will only assume for the moment that it is a subfunctor of some mini-functor $\wis{mini}(S)$ . That is, for every finite abelian group A we have an inclusion of sets $N(A) \subset Hom_{\Z-alg}(S,\Z A)$ in such a way that these inclusions are compatible with morphisms. Again, take pen and paper and you are bound to discover the correct definition of what is called a natural transformation, that is, a ‘map’ between the two functors $\wis{nano}(N) \rightarrow \wis{mini}(S)$ .

Right, now to make sense of our virtual F_un geometrical object $\wis{nano}(N)$ we have to connect it to properly existing complex- and/or integral-geometrical objects.

Let us define a gadget to be a couple $~(\wis{nano}(N),\wis{maxi}(R))$ consisting of a nano- and a maxi-functor together with a ‘map’ (that is, a natural transformation) between them

$e~:~\wis{nano}(N) \rightarrow \wis{maxi}(R)$

The idea of this map is that it visualizes the elements of the set as $\C A$ -points of the complex variety (that is, as a collection of points of , where is the number of elements of ).

In the example we used last time (the forgetful functor) with any group-element $a \in A$ is mapped to the algebra map $\C[x,x^{-1}] \rightarrow \C A~,~x \mapsto e_a$ in $\wis{maxi}(\C[x,x^{-1}])$ . On the geometry side, the points of the variety associated to $\C A$ are all algebra maps $\C A \rightarrow \C$ , that is, the characters $\{ \chi_1,\hdots,\chi_{o(A)} \}$ . Therefore, a group-element $a \in A$ is mapped to the $\C A$ -point of the complex variety $\C^* = X_{\C[x,x^{-1}]}$ consisting of all character-values at : $\{ \chi_1(a),\hdots,\chi_{o(A)}(g) \}$ .

In mathematics we do not merely consider objects (such as the gadgets defined just now), but also the morphisms between these objects. So, what might be a morphism between two gadgets

$~(\wis{nano}(N),\wis{maxi}(R)) \rightarrow (\wis{nano}(N'),\wis{maxi}(R'))$

Well, naturally it should be a ‘map’ (that is, a natural transformation) between the nano-functors $\phi~:~\wis{nano}(N) \rightarrow \wis{nano}(N')$ together with a morphism between the complex varieties $X_R \rightarrow X_{R'}$ (or equivalently, an algebra morphism $\psi~:~R' \rightarrow R$ ) such that the extra gadget-structure (the evaluation maps) are preserved.

That is, for every finite Abelian group we should have a commuting diagram of maps

$\xymatrix{N(A) \ar[rr]^{\phi(A)} \ar[d]^{e_N(A)} & & N'(A) \ar[d]^{e_{N'}(A)} \\ Hom_{\C-alg}(R,\C A) \ar[rr]^{- \circ \psi} & & Hom_{\C-alg}(R',\C A)}$

Not every gadget is a Fun variety though, for those should also have an integral form, that is, define a mini-functor. In fact, as we will see next time, an affine $\mathbb{F}_1$ -variety is a gadget determining a unique mini-functor $\wis{mini}(S)$ .