Theorem 2

Let M be an isotypical A-module of type S and $(V, \alpha )$ a description of M. Denote by $\mathscr{D}_D(V)$ the set, ordered by inclusion, of D-linear subspaces of V, and by $\mathscr{D}_A(M)$ that of A-submodules of M. For every $W\in$ $\mathscr{D}_D(V)$, identify the A-module $S\otimes_DW$ with its canonical image in $S\otimes_DV$.

a) The mapping $W\mapsto \alpha (S\otimes_DW)$ is an isomorphism of ordered sets from $\mathscr{D}_D(V)$ to $\mathscr{D}_A(M)$.

b) The inverse isomorphism sends a submodule N of M to the linear subspace of V consisting of the elements $v$ such that $\alpha (s\otimes v)$ belongs to N for every $s\in S$.

For $W\in \mathscr{D}_D$(V), set $\varphi (W) =\alpha (S\otimes_DW)$. For $N\in \mathscr{D}_A$(M), denote by $\psi (N)$ the set of elements $v\in V$ such that $\alpha (s\otimes v)\in N$ for every $s\in S$. This defines two mappings $\varphi :\mathscr{D}_D(V)\rightarrow \mathscr{D}_A(M)$ and $\psi :\mathscr{D}_A(M)\rightarrow \mathscr{D}_D(V)$. They are clearly increasing.

Let N be a submodule of M. It is isotypical of type S (VIII, p. 61, Proposition 2). Set $W =\psi (N)$. By Proposition 3, b) of VIII, p. 62, the A-linear mappings $h: S\rightarrow M$ are simply the mappings $s\mapsto \alpha (s\otimes v)$, where $v$ runs through V. Those with image contained in N are the mappings $s\mapsto \alpha (s\otimes w)$ where $w$ runs through W; their images generate N because N is isotypical of type S. We therefore have $\alpha (S\otimes_DW) = N$, that is, $\varphi (\psi (N)) = N$. This proves that $\varphi \circ \psi$ is the identity mapping on $\mathscr{D}_A(M)$. In particular, $\varphi$ is surjective and $\psi$ is injective.

To finish the proof, it suffices to prove that the mapping $\varphi$ is injective. Let W and $W'$ be linear subspaces of V such that $\varphi (W) =\varphi (W')$. The vector spaces $S\otimes_DW$ and $S\otimes_DW'$ coincide when viewed as linear subspaces of $S\otimes_DV$. Choose a nonzero linear form $f$ on the D-vector space S, and let $g: S\otimes_DV\rightarrow V$ be the group homomorphism defined by $g(s\otimes v) =f(s)v$. We have $W =g(S\otimes_DW) =g(S\otimes_DW') = W'$, so $\varphi$ is injective.