Today I’d like to show that the space of homomorphisms between two -modules is “additive”. That is, it satisfies the isomorphisms:
We should be careful here: the direct sums inside the are direct sums of -modules, while those outside are direct sums of vector spaces.
The second of these is actually the easier. If is a -morphism, then we can write it as , where and . Indeed, just take the projection and compose it with to get . These projections are also -morphisms, since and are -submodules. Since every can be uniquely decomposed, we get a linear map .
Then the general rules of direct sums tell us we can inject and back into , and write
Thus given any -morphisms and we can reconstruct an . This gives us a map in the other direction — — which is clearly the inverse of the first one, and thus establishes our isomorphism.
Now that we’ve established the second isomorphism, the first becomes clearer. Given a -morphism we need to find morphisms . Before we composed with projections, so this time let’s compose with injections! Indeed, composes with to give . On the other hand, given morphisms , we can use the projections and compose them with the to get two morphisms . Adding them together gives a single morphism, and if the came from an , then this reconstructs the original. Indeed:
And so the first isomorphism holds as well.
We should note that these are not just isomorphisms, but “natural” isomorphisms. That the construction is a functor is clear, and it’s straightforward to verify that these isomorphisms are natural for those who are interested in the category-theoretic details.