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Connes-Consani for undergraduates (2)

Posted By lievenlb On October 8, 2008 @ 8:35 pm In Connes-Consani2008,undergraduate | No Comments

Last time [1] 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 F_un 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 S_n 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 S_n 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 N(A) as \C A-points of the complex variety X_R (that is, as a collection of o(A) points of X_R, where o(A) is the number of elements of A).

In the example we used last time [1] (the forgetful functor) with N(A)=A 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 o(A) 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 a : \{ \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 A 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 F_un 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).


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[1] Last time: http://cage.ugent.be/~kthas/Fun/index.php/connes-consani-for-undergraduates-1.html

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