Connes-Consani for undergraduates (2)
Posted by lievenlb on Oct 8, 2008 in Connes-Consani2008, undergraduate • No commentsLast time 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 (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
.
