Remark 1

Let $\mathscr{S}\mathscr{S}(A)$ be the hereditary set of classes of semisimple A-modules of finite length. By Proposition 7 of VIII, p. 190, the Grothendieck group $K(\mathscr{S}\mathscr{S}(A))$ is a free $\mathbf{Z}$-module whose elements $[S]_{\mathscr{S}\mathscr{S}(A)}$ (for $S\in \mathscr{S}(A)$) form a basis. The canonical homomorphism $\gamma_{\mathscr{L}\mathscr{F}(A),\mathscr{S}\mathscr{S}(A)}$ (VIII, p. 186) is therefore an isomorphism.