Remark 2

There exists a homomorphism $\varphi$ from $R_K(A_1)\otimes_{\mathbf{Z}}R_K(A_2)$ to $R_K(A)$ characterized by the relation $\varphi ([M_1]\otimes [M_2]) = [M_1\otimes M_2]$. This can be proved in the same way as Proposition 9 of VIII, p. 196. If the field K is algebraically closed, then $\varphi$ is an isomorphism from $R_K(A_1)\otimes_ZR_K(A_2)$ to $R_K(A)$ by Theorem 1, b) because for every K-algebra B, the $\mathbf{Z}$-module $R_K(B)$ is free with basis the family $([S])_{S\in\mathscr{S}_K(B)}$ (VIII, p. 195).