Jacobson's Density Theorem for Semisimple Algebras

Question

I am trying to follow the proof of the following:

Let $A$ be an algebra over an algebraically closed field $k$. Let $V=V_1\oplus \ldots \oplus V_r$, where $V_i$ are pairwise non-isomorphic irreducible finitely dimensional representations of $A$. Then the map $\oplus_{i=1}^r\rho_i: A\to \oplus_{i=1}^r \operatorname{End}_k(V_i)$ is surjective.

I am working out of Etingof's text (Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina, Introduction to representation theory, 10 January 2011) (Theorem 2.5 page 24 in the link and if you have a hard copy of the text this is Theorem 3.2.2 on page 43). The proof is not too bad but there is one detail that I'm having difficulties with.

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My issue is he quotes Proposition 2.2 (from the link above) to conclude that Im$\oplus_{i=1}^r \rho_i=\oplus_{i=1}^r$Im$\rho_i$. I am having difficulties seeing how this follows from the proposition. The rest of the proof and material leading up to this is fine. Any help would be greatly appreciated.

Thanks.

Answer 1

$\newcommand{\bop}{\bigoplus\limits}$ $\newcommand{\End}{\operatorname*{End}}$ The image $B$ of the map $\bop_{i=1}^r \rho_i : A \to \bop_{i=1}^r \End\left(V_i\right)$ is an $A$-submodule of $\bop_{i=1}^r \End\left(V_i\right)$ (since $\bop_{i=1}^r \rho_i$ is an $A$-module homomorphism), and thus (by Proposition 2.2 -- actually by the very first statement of Proposition 2.2) is a direct sum $\bop_{i=1}^r W_i$ of $A$-submodules $W_i$ of $\End V_i$ (since each $\End\left(V_i\right)$ is a direct sum of copies of $V_i$, and since all $V_i$ are irreducible and pairwise distinct). Thus, for any fixed $1 \leq j \leq r$, the composition $\pi_j \circ \left(\bop_{i=1}^r \rho_i\right)$ of this map $\bop_{i=1}^r \rho_i$ with the projection $\pi_j : \bop_{i=1}^r \End\left(V_i\right) \to \End\left(V_j\right)$ must have image $W_j$. But this composition is simply $\rho_j$ and thus has image $\End\left(V_j\right)$ (since Theorem 2.5 (i) yields that the map $\rho_j$ is surjective). Thus, we obtain $W_j = \End\left(V_j\right)$. In other words, $W_i = \End\left(V_i\right)$ for every $1 \leq i \leq r$. Hence, the image of $\bop_{i=1}^r \rho_i$ is $B = \bop_{i=1}^r \underbrace{W_i}_{=\End\left(V_i\right)} = \bop_{i=1}^r \End\left(V_i\right)$. In other words, $\bop_{i=1}^r \rho_i$ is surjective.