Remark 3

Suppose that P is finitely generated as an A-module. Let M be the direct sum of a family $(M_i)_{i\in I}$ of A-modules. The A-modules $\mathscr{T}(\mathscr{H}(M))$ and $\bigoplus_i\mathscr{T}(\mathscr{H}(M_i))$ are canonically isomorphic. When we identify them, $\alpha_M$ is identified with $\bigoplus_i\alpha_{M_i}$. Likewise, let V be the direct sum of a family $(V_j)_{j\in J}$ of B-modules. The B-module $\mathscr{H}(\mathscr{T}(V))$ is identified with $\bigoplus_j\mathscr{H}(\mathscr{T}(V_j))$, and the linear mapping $\beta_V$ with $\bigoplus_j\beta_{V_j}$.