Theorem 3

Let M and $M'$ be isotypical A-modules of type S. Let $(V, \alpha )$ and $(V', \alpha ')$ be descriptions of M and $M'$, respectively. For any D-linear mapping $f: V\rightarrow V'$, denote by $\widetilde{f}: M\rightarrow M'$ the unique A-linear mapping that makes the following diagram commute:

$S\otimes_DV^{\alpha}$ // M

$S\otimes_D{V'}^{\alpha'}$ // $M'$.

The mapping $f\mapsto \widetilde{f}$ from Hom$_D(V,V')$ to Hom$_A(M,M')$ is a group isomorphism.

It suffices to prove that the $\mathbf{Z}$-linear mapping $u\mapsto 1_S\otimes u$ from Hom$_D(V,V')$ to Hom$_A(S\otimes_DV,S\otimes_DV')$ is bijective. In the notation of No.3 applied to the $(A$, D)-bimodule S, this corresponds to showing that the mapping

$\mathscr{T}:$ Hom$_D(V,V')\longrightarrow$ Hom$_A(\mathscr{T}(V),\mathscr{T}(V'))$

is bijective. But by Remark 1 of VIII, p. 60, since the adjunction isomorphism (VIII, p. 59) is bijective, this corresponds to showing that the mapping that sends $u$ to $\beta_{V'}\circ u$ is bijective, which follows from the fact that the D-linear mapping $\beta_{V'}$ is bijective (VIII, p. 62, Proposition 3, b)).

We keep the notation of Theorem 3. Let $M''$ be an isotypical A-module of type S, and let $(V'', \alpha '')$ be a description of $M''$. For every $f\in$ Hom$_D(V,V')$ and every $g\in$ Hom$_D(V',V'')$, we have $g]\circ f=\widetilde{g}\circ \widetilde{f}$. In particular, for $M = M'$, $V = V'$, and $\alpha =\alpha '$, the mapping $f\mapsto \widetilde{f}$ from End$_D(V)$ to End$_A(M)$ is a ring isomorphism.