Connes-Consani for undergraduates (2)

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

$\mathbf{maxi}(R):\mathbf{abelian}\to \mathbf{sets}A\mapsto Ho{m}_{\mathbb{C}-alg}(R,\mathbb{C}A)$

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

$\mathbf{mini}(S):\mathbf{abelian}\to \mathbf{sets}A\mapsto Ho{m}_{\mathbb{Z}-alg}(S,\mathbb{Z}A)$

Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety $X_R=\mathbf{max}(R)$ (the set of maximal ideals of the algebra R) with complex coordinate ring R and for a mini-functor by the integral affine scheme $X_S=\mathbf{spec}(S)$ (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 ${\mathbb{C}}^n$ has as its integral form a lattice of rank n ${\mathbb{Z}}^{\oplus n}$. 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 ${\mathbb{C}}^n$ are given by invertible matrices with complex coefficients $G{L}_n(\mathbb{C})$. Of these base-changes, the only ones leaving the integral lattice ${\mathbb{Z}}^{\oplus n}$ intact are the matrices having all their entries integers and their determinant equal to $\pm 1$, that is the group $G{L}_n(\mathbb{Z})$. 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 $G{L}_n(\mathbb{C})$.

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

$\mathbf{nano}(N):\mathbf{abelian}\to \mathbf{sets}A\mapsto N(A)$

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 $\mathbf{mini}(S)$. That is, for every finite abelian group A we have an inclusion of sets $N(A)\subset Ho{m}_{\mathbb{Z}-alg}(S,\mathbb{Z}A)$ 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 $\mathbf{nano}(N)\to \mathbf{mini}(S)$.

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

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

$e:\mathbf{nano}(N)\to \mathbf{maxi}(R)$

The idea of this map is that it visualizes the elements of the set $N(A)$ as $\mathbb{C}A$-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 (the forgetful functor) with $N(A)=A$ any group-element $a\in A$ is mapped to the algebra map $\mathbb{C}[x,x^{-1}]\to \mathbb{C}A,x\mapsto e_a$ in $\mathbf{maxi}(\mathbb{C}[x,x^{-1}])$. On the geometry side, the points of the variety associated to $\mathbb{C}A$ are all algebra maps $\mathbb{C}A\to \mathbb{C}$, that is, the $o(A)$ characters ${\chi }_1,\dots ,{\chi }_{o(A)}$. Therefore, a group-element $a\in A$ is mapped to the $\mathbb{C}A$-point of the complex variety ${\mathbb{C}}^{\ast }=X_{\mathbb{C}[x,x^{-1}]}$ consisting of all character-values at $a$ : ${\chi }_1(a),\dots ,{\chi }_{o(A)}(g)$.

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

$(\mathbf{nano}(N),\mathbf{maxi}(R))\to (\mathbf{nano}(N^{\prime }),\mathbf{maxi}(R^{\prime }))$

Well, naturally it should be a ‘map’ (that is, a natural transformation) between the nano-functors $\varphi :\mathbf{nano}(N)\to \mathbf{nano}(N^{\prime })$ together with a morphism between the complex varieties $X_R\to X_{R^{\prime }}$ (or equivalently, an algebra morphism $\psi :R^{\prime }\to R$) 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

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 $\mathbf{mini}(S)$.