The Unapologetic Mathematician

Hom-Space Additivity

Today I’d like to show that the space $\hom_G(V,W)$ of homomorphisms between two $G$-modules is “additive”. That is, it satisfies the isomorphisms:

\displaystyle\begin{aligned}\hom_G(V_1\oplus V_2,W)&\cong\hom_G(V_1,W)\oplus\hom_G(V_2,W)\\\hom_G(V,W_1\oplus W_2)&\cong\hom_G(V,W_1)\oplus\hom_G(V,W_2)\end{aligned}

We should be careful here: the direct sums inside the $\hom$ are direct sums of $G$-modules, while those outside are direct sums of vector spaces.

The second of these is actually the easier. If $f:V\to W_1\oplus W_2$ is a $G$-morphism, then we can write it as $f=(f_1,f_2)$, where $f_1:V\to W_1$ and $f_2:V\to W_2$. Indeed, just take the projection $\pi_i:W_1\oplus W_2\to W_i$ and compose it with $f$ to get $f_i=\pi_i\circ f$. These projections are also $G$-morphisms, since $W_1$ and $W_2$ are $G$-submodules. Since every $f$ can be uniquely decomposed, we get a linear map $\hom_G(V,W_1\oplus W_2)\to\hom_G(V,W_1)\oplus\hom_G(V,W_2)$.

Then the general rules of direct sums tell us we can inject $W_1$ and $W_2$ back into $W_1\oplus W_2$, and write

$\displaystyle f=I_{W_1\oplus W_2}\circ f=(\iota_1\circ\pi_1+\iota_2\circ\pi_2)\circ f=\iota_1\circ f_1+\iota_2\circ f_2$

Thus given any $G$-morphisms $f_1:V\to W_1$ and $f_2:V\to W_2$ we can reconstruct an $f:V\to W_1\oplus W_2$. This gives us a map in the other direction — $\hom_G(V,W_1)\oplus\hom_G(V,W_2)\to\hom_G(V,W_1\oplus W_2)$ — 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 $G$-morphism $h:V_1\oplus V_2\to W$ we need to find morphisms $h_i:V_i\to W$. Before we composed with projections, so this time let’s compose with injections! Indeed, $\iota_i:V_i\to V_1\oplus V_2$ composes with $h$ to give $h_i=h\circ\iota_i:V_i\to W$. On the other hand, given morphisms $h_i:V_i\to W$, we can use the projections $\pi_i:V_1\oplus V_2\to V_i$ and compose them with the $h_i$ to get two morphisms $h_i\circ\pi_i:V_1\oplus V_2\to W$. Adding them together gives a single morphism, and if the $h_i$ came from an $h$, then this reconstructs the original. Indeed:

$\displaystyle h_1\circ\pi_1+h_2\circ\pi_2=h\circ\iota_1\circ\pi_1+h\circ\iota_2\circ\pi_2=h\circ(\iota_1\circ\pi_1+\iota_2\circ\pi_2)=h\circ I_{V_1\oplus V_2}=h$

And so the first isomorphism holds as well.

We should note that these are not just isomorphisms, but “natural” isomorphisms. That the construction $\hom_G(\underline{\hphantom{X}},\underline{\hphantom{X}})$ 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 -

3 Comments »

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