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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 -algebra R gives us a **maxi-functor** (because the associated sets are typically huge)

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

Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety (the set of maximal ideals of the algebra R) with complex coordinate ring R and for a mini-functor by the integral affine scheme (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 records the essence of possibly complicated complex- or integral- objects in a small combinatorial thing.

For example, an n-dimensional complex vectorspace has as its integral form a lattice of rank n . The corresponding -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 are given by invertible matrices with complex coefficients . Of these base-changes, the only ones leaving the integral lattice intact are the matrices having all their entries integers and their determinant equal to , that is the group . 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 : the finite group is the Weyl-group of the complex Lie group .

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

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 . That is, for every finite abelian group A we have an inclusion of sets 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 .

Right, now to make sense of our virtual F_un geometrical object we have to connect it to properly existing complex- and/or integral-geometrical objects.

Let us define a **gadget** to be a couple consisting of a nano- and a maxi-functor together with a ‘map’ (that is, a natural transformation) between them

The idea of this map is that it visualizes the elements of the set as -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 ^{[1]} (the forgetful functor) with any group-element is mapped to the algebra map in . On the geometry side, the points of the variety associated to are all algebra maps , that is, the **characters** . Therefore, a group-element is mapped to the -point of the complex variety consisting of all character-values at : .

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

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

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 -variety is a gadget determining a unique mini-functor .

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