Proposition 7

The canonical mapping (23) is bijective.

To prove this it suffices to define a $\mathbf{Z}$-linear mapping $h$ of the direct sum $G = \bigoplus_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu)$ into $E \otimes_A F$ such that $g \circ h$ and $h \circ g$ are the identity mappings. But, to define a $\mathbf{Z}$-linear mapping of $G$ into $E \otimes_A F$, it suffices (§ 1, no. 6, Proposition 6) to define a $\mathbf{Z}$-linear mapping

for every ordered pair $(\lambda, \mu)$, and we take $h_{\lambda \mu} = i_\lambda \otimes j_\mu$, where $i_\lambda : E_\lambda \to E$ and $j_\mu : F_\mu \to F$ are the canonical injections. Then clearly $h \circ g$ coincides with the identity mapping for the elements of the form $\left( \sum_\lambda x_\lambda \right) \otimes \left( \sum_\mu y_\mu \right)$ which generate the $\mathbf{Z}$-module $E \otimes_A F$; similarly $g \circ h$ coincides with the identity mapping for the elements of the form $\sum_{\lambda, \mu} (x_\lambda \otimes y_\mu)$ which generate the $\mathbf{Z}$-module $G$, since for each ordered pair $(\lambda, \mu)$ the products

generate the $\mathbf{Z}$-module $E_\lambda \otimes_A F_\mu$. Whence the proposition.

Let $u_\lambda : E_\lambda \to E'_\lambda$, $v_\mu : F_\mu \to F'_\mu$ be $A$-homomorphisms; clearly the diagram

is commutative.