Proposition 5

a) The countermodule of M is semisimple.

b) For every $\lambda \in \mathscr{S}_M$, the B-module $V_{\lambda}$ is simple, and its commutant is equal to $(D_{\lambda})_{V_{\lambda}}$.

c) The mapping $\lambda \mapsto$ cl(V$_{\lambda}$) is a bijection from the support of the A-module M to the support of its countermodule.

d) For every $\lambda \in \mathscr{S}_M$, the B-submodule $M_{\lambda}$ is the isotypical component of type $V_{\lambda}$ of the B-module M, and the multiplicity of $V_{\lambda}$ in M is equal to dim$_{D_{\lambda}}(S_{\lambda})$.

e) For $s\in S$, denote the mapping $\varphi \mapsto \varphi (s)$ from $V_{\lambda}=$ Hom$_A(S_{\lambda},M)$ to M by $\widetilde{s}$. It is B-linear. The resulting mapping $s\mapsto \widetilde{s}$ from $S_{\lambda}$ to Hom$_B(V_{\lambda},M)$ is an isomorphism of $(A,D_{\lambda})$-bimodules.

Let $\lambda \in \mathscr{S}_M$. Denote the ring End$_{D_{\lambda}}(V_{\lambda})$ by $E_{\lambda}$; since $V_{\lambda}$ is a nonzero $D_{\lambda}$-vector space, it is a simple $E_{\lambda}$-module, (VIII, p. 45, Example 3), and its commutant is equal to $(D_{\lambda})_{V_{\lambda}}$ (VIII, p. 82, Corollary 1 of Theorem 2). Since $E_{\lambda}$ is the ring of homotheties of the B-module $V_{\lambda}$ (VIII, p. 71, Corollary of Proposition 9), this proves b).

The canonical description $\alpha_M$ of M defines an isomorphism $\alpha_{\lambda}$ from $V_{\lambda}\otimes_{D^o_{\lambda}}S_{\lambda}$ to $M_{\lambda}$. Since $V_{\lambda}$ is a simple B-module, the B-module $V_{\lambda}\otimes_{D^o_{\lambda}}S_{\lambda}$ is isotypical of type $V_{\lambda}$ (VIII, p. 61, Proposition 1); the same therefore holds for the B-module $M_{\lambda}$, which proves a).

By Remark 1 above, there exist elements $e_{\lambda}$ of B, for $\lambda$ running through $\mathscr{S}_M$, such that $(e_{\lambda})_{V_{\lambda}}= 1_{V_{\lambda}}$ and $(e_{\lambda})_{V_\mu}$ = 0 for $\mu\in \mathscr{S}_M$ with $\mu\not=\lambda$. The simple B-modules $V_{\lambda}$ are therefore pairwise nonisomorphic, which proves c) and the first assertion of d). The B-module $M_{\lambda}$ is isomorphic to $V_{\lambda}\otimes_{D^o_{\lambda}}S_{\lambda}$, so dim$_{D_{\lambda}}(S_{\lambda})$ is the multiplicity of $V_{\lambda}$ in M (II, §3, No. 7, p. 255, Corollary 1).

The mapping $\sum_{\lambda\in\mathscr{S}_M}\alpha_{\lambda}$ from $\bigoplus_{\lambda\in\mathscr{S}_M}V_{\lambda}\otimes_{D^o_{\lambda}}S_{\lambda}$ to M provides a description (VIII, p. 69, Definition 5) of the semisimple B-module M. By VIII, p. 70, Proposition 8, b), for every $\lambda \in \mathscr{S}_M$, the mapping from $S_{\lambda}$ to Hom$_B(V_{\lambda},M)$ described in e) is bijective and $D_{\lambda}$-linear. Since it is obviously A-linear, this proves e).