Lemma 1

Let $M_1$ and $N_1$ be $A_1$-modules, and let $M_2$ and $N_2$ be $A_2$-modules. We make the following assumptions:

(i) The $A_1$-module $M_1$ is finitely generated.

(ii) The $A_2$-module $M_2$ is finitely generated, or $N_1$ is finite-dimensional over K.

Set $M = M_1\otimes M_2$ and $N = N_1\otimes N_2$, and view them as modules over the ring $A = A_1\otimes A_2$. The canonical homomorphism (II, §3, No. 5, p. 251)

$\lambda :$ Hom$_K(M_1,N_1)\otimes$ Hom$_K(M_2,N_2)\longrightarrow$ Hom$_K(M,N)$

then induces an isomorphism of K-vector spaces

$\varphi :$ Hom$_{A_1}(M_1,N_1)\otimes$ Hom$_{A_2}(M_2,N_2)\longrightarrow$ Hom$_A(M,N)$.

The mapping $\lambda$ is injective (II, §7, No. 7, p. 308, Proposition 16) and sends the linear subspace Hom$_{A_1}(M_1,N_1)\otimes$ Hom$_{A_2}(M_2,N_2)$ to Hom$_A(M,N)$. It therefore suffices to prove that every A-linear mapping from M to N belongs to the image of Hom$_{A_1}(M_1,N_1)\otimes$ Hom$_{A_2}(M_2,N_2)$ by $\lambda$. Let $u: M\rightarrow N$ be an A-linear mapping. Let $x\in M_1$. Denote by $u_x$ the $A_2$-linear mapping $y\mapsto u(x\otimes y)$ from $M_2$ to $N_1\otimes N_2$. Set P = Hom$_{A_2}(M_2,N_2)$, and denote by $\nu$ the canonical homomorphism from $N_1\otimes P$ to Hom$_{A_2}(M_2,N_1\otimes N_2)$ (II, §4, No. 2, p. 269). This mapping is injective (II, §4, No. 2, p. 269, Proposition 2, (i) applied to the K-vector space $N_1$). By assumption (ii), there exists a linear subspace $V_x$ of $N_1$, finite-dimensional over K, such that $u_x$ takes on values in $V_x\otimes N_2$. It follows that $u_x$ is the image by $\nu$ of a unique element $v_x$ of $N_1\otimes P$. The mapping $\widetilde{u}:x\mapsto v_x$ from $M_1$ to $N_1\otimes P$ is $A_1$-linear. By assumption (i), the $A_1$-module $M_1$ is finitely generated. A reasoning analogous to the above shows that $\widetilde{u}$ belongs to the image of Hom$_{A_1}(M_1,N_1)\otimes P$ in Hom$_{A_1}(M_1,N_1\otimes P)$. Lemma 1 follows.