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Connes-Consani for undergraduates (2)

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

maxi(R) : abeliansetsAHomCalg(R,CA)

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

mini(S) : abeliansetsAHomZalg(S,ZA)

Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety XR=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 XS=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 F1 records the essence of possibly complicated complex- or integral- objects in a small combinatorial thing.

For example, an n-dimensional complex vectorspace Cn has as its integral form a lattice of rank n Zn. The corresponding F1-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 Cn are given by invertible matrices with complex coefficients GLn(C). Of these base-changes, the only ones leaving the integral lattice Zn intact are the matrices having all their entries integers and their determinant equal to ±1, that is the group GLn(Z). Of these integral matrices, the only ones that shuffle the base-vectors around are the permutation matrices, that is the group Sn of all possible ways to permute the n base-vectors. In fact, this example also illustrates Tits’ original motivation to introduce F1 : the finite group Sn is the Weyl-group of the complex Lie group GLn(C).

So, we expect a geometric F1-object to determine a much smaller functor from finite abelian groups to sets, and, therefore we call it a nano-functor

nano(N) : abeliansetsAN(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 mini(S). That is, for every finite abelian group A we have an inclusion of sets N(A)HomZalg(S,ZA) 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 nano(N)mini(S).

Right, now to make sense of our virtual F_un geometrical object 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  (nano(N),maxi(R)) consisting of a nano- and a maxi-functor together with a ‘map’ (that is, a natural transformation) between them

e : nano(N)maxi(R)

The idea of this map is that it visualizes the elements of the set N(A) as CA-points of the complex variety XR (that is, as a collection of o(A) points of XR, 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 aA is mapped to the algebra map C[x,x1]CA , xea in maxi(C[x,x1]). On the geometry side, the points of the variety associated to CA are all algebra maps CAC, that is, the o(A) characters χ1,,χo(A). Therefore, a group-element aA is mapped to the CA-point of the complex variety C=XC[x,x1] consisting of all character-values at a : χ1(a),,χo(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

 (nano(N),maxi(R))(nano(N),maxi(R))

Well, naturally it should be a ‘map’ (that is, a natural transformation) between the nano-functors ϕ : nano(N)nano(N) together with a morphism between the complex varieties XRXR (or equivalently, an algebra morphism ψ : RR) 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

Misplaced &

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 F1-variety is a gadget determining a unique mini-functor mini(S).

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