Projective Localization, Exterior Power

Exterior Power

Like tensor product, the exterior power can be defined in two ways, as a universal object in a carefully constructed category, and as the quotient of the cross product of modules mod certain relations. Naturally these two definitions will coincide. You need to be familiar with the tensor product and bilinear/multilinear functions before you proceed.

Let R be commutative and let M be an R module. Let j be a fixed integer, indicating the j^th exterior power of M. Let W be the direct product M^j. These are j-tuples of elements drawn from M.

A function f from W into some other R module P is multilinear if it respects addition per component, and the action of R can be passed between components. If d is in R, and x and y are values of M for 2 of the j components, then f(…x,dy…) = f(…dx,y…), where the other j-2 components are fixed. Note that f is not an R module homomorphism, or even a group homomorphism. It respects addition per component, but does not respect addition across the board. That is, f(2,2,2) need not equal f(1,1,1) + f(1,1,1); but f(7,1,1) definitely equals f(3,1,1) + f(4,1,1).

The function f is symmetric multilinear if the image of f is 0 whenever any two of the components of W are equal. Thus f(2,7,3) has to be the opposite of f(5,7,3).

This has the effect of making tuples order invariant, up to sign. In the following equation, * is the rest of the tuple.

0 =
f(9,9,*) =
( f(4,4,*) + f(4,5,*) ) + ( f(5,4,*) + f(5,5,*) ) =
f(4,5,*) + f(5,4,*)

As shown by the last line, swapping two components negates the sign of f. In general, permuting the components multiplies f by ±1, according to the parity of the permutation.

Consider the category of symmetric multilinear functions from W into abelian groups. A morphism from the abelian group P into Q is a group homomorphism that is compatible with the two maps from W into P and from W into Q. The exterior power E, with its attendant function f from W onto E, is universal in this category. What does this mean? If g is a symmetric multilinear function from W into P, There is a unique homomorphism h from E into P such that fh = g.

The exterior product is unique up to isomorphism, using the proof shown in the tensor product.

Let T be the tensor product of j coppies of M. Since f(W) → E is a multilinear map, there is a unique, compatible function from T onto E. In other words, E is the quotient of T, the image of the tensor product. Let's use this to build E.

Let K be a kernel of T, generated by those tuples that exhibit repeated elements. Let E be the quotient group T/K. Let f be the composition of the tensor function from W onto T, with the quotient map from T onto E.

We need to prove E is universal. Let g be a symmetric multilinear function from W into P. Take advantage of the fact that T is universal, and let j be the unique homomorphism from T into P that is compatible with g. To agree with g, j has to be 0 on the kernel K. Thus j induces a well defined map h from E into P that is compatible with g. I'm skating past a lot of details, but I think it's pretty intuitive.

R module

As you recall, the tensor product was created as an abelian group, yet it turns out to be an R module, and the function from W onto T is an R homomorphism. In the same way, E is an R module (since K is a submodule of T), and f(W) onto E is an R module homomorphism. Also, E is universal in the category of symmetric R multilinear functions from W into various R modules. This mirrors what we have done before with the tensor product.

Generators

If M is spanned by a certain set of generators, tuples of these generators combine to generate T, and then E. Relations in the presentation of M appear, per component, and crossed with the other generators in the other components, to create the presentation of the tensor product T. Fold in the relations of K, and you have built E.

Assume the generators of M can be linearly ordered. (If you accept the axiom of choice, they can always be well ordered.) Combine j of these generators to create a generator for E. However, if the second component is smaller than the first, i.e. out of order, we can swap them. The two tuples are opposite, and we don't need them both. In fact, we only need tuples where the generators are in ascending order. This will help us analyze the exterior product of a free module, in the next section.

If M is spanned by fewer than j generators, we cannot place j generators in ascending order; hence E = 0.