Remark

Let $\mathscr{S}_K$ be the set of classes of simple A-modules of finite dimension over K. Proposition 6 also follows from the fact that the sum of the $\Theta_S$(A), for S running through $\mathscr{S}_K$, is direct.

Let

be an exact sequence of A-modules, all finite-dimensional over K. By Proposition 1 of III, §9, No. 2, p. 543, we have Tr$_E=$ Tr$_{E'}+$ Tr$_{E''}$. By the definition of the Grothendieck group $R_K(A)$ (VIII, p. 194) and its universal property (VIII, p. 186, Proposition 4), there exists a homomorphism of additive groups $\theta$ from $R_K(A)$ to $A^*$ characterized by the relation $\theta ([E]) =$ Tr$_E$ for every A-module E of finite dimension over K. In particular, the trace of a semisimplification of E is equal to that of E. We deduce from $\theta$ a K-linear mapping $\theta_K: K\otimes_{\mathbf{Z}}R_K(A)\rightarrow A^*$.