Semisimplicity of End$_{F[G]}(M)$

Question

Let $F$ be a field of characteristic $0$ and $G$ be a finite group. Let $M$ be a finitely generated free $F[G]$-module where $F[G]$ is the usual group ring.

Can anyone explain to me why the $F$-algebra End$_{F[G]}(M)$ is semisimple?

Answer 1

By Maschke's theorem, $F[G]$ is a semisimple ring, and so $M$ is a semisimple $F[G]$ module. Since it is finitely generated, it is a direct sum of finitely many simple submodules.

If you know the ideas behind the Artin-Wedderburn theorem well, then you can derive that $M$ breaks into pieces, which breaks the endomorphism ring of $M$ into pieces. Since $M$ is finitely generated, each of the pieces that $End_{F[G]}(M)$ breaks into are simple Artinian rings.

(An alternative high level explanation is that the endomorphism ring of a semisimple module is always von Neumann regular, and if the module is f.g. then it is Artinian too, whence it is semisimple. But please keep reading if this blurb is not useful!)

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More details about the ideas:

By basic ideas from the proof of the Artin-Wedderburn theorem, you can decompose $M$ into a direct sum of submodules which are "homogeneous." Explicitly, $M_S:=\oplus\{N<M\mid N\cong S\}$ where $S$ is a fixed simple submodule. Then $M=\oplus M_{S_k}$ for a representative set of simple submodules of $M$ $S_1,\dots S_n$.

Let's abbreviate $M_{S_k}$ to just $M_k$. Now if you look at $End_{F[G]}(M)$ through this decomposition, it takes the form of matrices of transformations whose $i,j$ entries are elements of $Hom(M_i,M_j)$. But it's easy to see that when $i\neq j$, the only homomorphism is zero, and so the only nonzero entries occur on the diagonal spots $Hom(M_i,M_i)$, which come from the endomorphism ring of $M_i$. So, $End_{F[G]}(M)=\oplus End_{F[G]}(M_i)$. If you show that each summand is semisimple, then you'll have shown the sum is semisimple.

Again, basic ideas of Artin-Wedderburn (or else you can skip them, if you already know them well) tell us that $End_{F[G]}(M_i)$ is isomorphic to $M_n(D)$ where $D=End(S_i)$ is a division ring and $S_i$ is the isotype of the simple submodules of $M_i$, and $n$ is the length of $M_i$. So, $End_{F[G]}(M_i)$ are all semisimple, and so is the sum of these rings.