Interactions between hom, tensor product, and direct sum
We now have three ways of putting modules together: the abelian group of left -module homomorphisms, the tensor product of a right -module and a left -module , and the direct sum of two left -modules. Today we consider their interactions.
First off, the universal property of direct sums tells us that a homomorphism is the same as a pair of homomorphisms, and . Given the first we can compose it with the projections to get the second, and given the second we can use the universal property to get the first. That is, we can make homomorphisms of abelian groups
and we can check that these are the injections and projections of a direct sum of abelian groups! That is:
where the direct sum on the left is of left -modules, while the one on the right is of abelian groups. Similarly, we can show that
and that these both work for all finite direct sums.
For infinite direct sums it’s a little trickier. An infinite direct sum in the second variable works just the same:
but if it shows up in the first variable we have to use the direct product of abelian groups to get the right universal properties to go through. Try following the above argument yourself to see where the difference is.
Tensor products and direct sums are similar, and don’t even have the same difficulties with infinite sums. After playing with universal properties of direct sums like we did above, we find that
Where things get really fun, though, is with tensor products and homs. Let’s consider a right -module , a left -module , and an abelian group . The universal property of tensor products tells us that a linear function from to is the same thing as a middle-linear function from to . Let’s consider such a middle linear function .
If we pick an element and stick it in the first slot of , we get . This is a linear function from to , so . Notice that we’re using for abelian groups, so has an extra left -module structure. It gets flipped over, turning into a right -module. Now, building out of the element is a homomorphism of right -modules. That is, given we can build an element of .
Even better, any such homomorphism gives rise to a middle-linear function from to . That is, we have an isomorphism:
For bonus points, go back through these interactions and try adding extra module structures to each of the modules we used.