Hom-Space Additivity
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.
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October 11, 2010 -
Posted by John Armstrong |
Algebra, Category theory, Group theory, Representation Theory
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