Some examples of modules
Today I want to run through a bunch of examples of the constructions we’ve been considering for modules. I’ll restrict to the case of a ring with unit.
One easy example of an -module that I’ve mentioned before is the ring itself. We drop down to the underlying abelian group and then act on it using the ring multiplication. There are both left and right actions here: and where and are ring elements, considered as an element of the module. We’ll start off by taking this module and sticking it into some of the constructions.
When we consider for some left -module the left module structures on and will get eaten and the right module structure on will get flipped over, leaving us a left -module. We can pick an element by specifying . Then , telling us where everything else goes. If we write for the homomorphism with , then the left action of on homomorphisms says
Thus . This means that as left -modules.
On the other hand, if we consider we get a right -module. This consists of all -linear functions from to the ring itself. We call this the “dual” module to , and write . Elements of the dual module are often called “linear functionals” on .
Tensor products are even easier. When we consider for a left -module we can use the construction of tensor products to write an element as a finite sum: . But then we can use the middle-linear property to write , and then the linearity to collect all the terms together, giving . The tensor product eats the module structure on and the right module structure on , leaving a left -module structure. We calculate
so as left -modules.
Now let’s take two left -modules and and make . This is an abelian group — a -module — as is . Let’s write as as above and then tensor over with . Then we can compose homomorphisms
This is the “evaluation” homomorphism that takes an element and a homomorphism and gives back .
As a special case, we can take itself in place of . We get an evaluation homomorphism . This “canonical pairing” we often write as for a linear functional and module element .
What if we composed with an element of instead of ? We use the evaluation homomorphism to get
So given a homomorphism we get a homomorphism
Of course, all this goes through suitably changed by swapping “right” for “left”. For example, given a right -module we have a dual left -module .
What do we get if we start with a left module , dualize it, then dualize again to get another left module ? Following the definitions we see . I claim that there is a natural morphism of left -modules . That is, a special element of
but we know that this is isomorphic to
which we write as
so we’re really looking for a special homomorphism from to . And we’ve got one: the canonical pairing! So we take the canonical pairing as a homomorphism from and pass it through this natural isomorphism to get a homomorphism . In case this looks completely insane, here it is in terms of elements: takes a linear functional and gives back an element of the ring by the rule .