Proposition 7

The $\mathbf{Z}$-linear mapping (7) is bijective.

We set $P = \lim (E_\alpha \otimes_{A_\alpha} F_\alpha)$ and, for all $\alpha \in I$, let $h_\alpha : E_\alpha \otimes_{A_\alpha} F_\alpha \to P$ be the canonical mapping. On the other hand, for all $\alpha \in I$, let

be the canonical $\mathbf{Z}$-bilinear mapping; for $\alpha \leq \beta$, and hence $(t_\alpha)$ is a direct system of mappings. Canonically identifying $\lim (E_\alpha \times F_\alpha)$ with $E \times F$ (Set Theory, III, § 7, no. 7, Proposition 10), we derive a mapping $t = \lim t_\alpha : E \times F \to P$ such that Taking account of Set Theory, III, § 7, no. 5, Lemma 1, it is immediately seen that $t$ is $\mathbf{Z}$-bilinear; moreover, for $x \in E, y \in F, \lambda \in A$, there exists $\alpha \in I$ such that $x = f_\alpha(x_\alpha), y = g_\alpha(y_\alpha), \lambda = \rho_\alpha(\lambda_\alpha)$ with $\lambda_\alpha \in A_\alpha, x_\alpha \in E_\alpha, y_\alpha \in F_\alpha$ (Set Theory, III, § 3, no. 7, Lemma 1); whence Hence there exists one and only one $\mathbf{Z}$-linear mapping $\pi' : E \otimes_A F \to P$ such that $\pi'(x \otimes y) = t(x, y)$ (§ 3, no. 1, Proposition 1). Moreover, by definition and as the elements of the form $f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha)$ (resp. $h_\alpha(x_\alpha \otimes y_\alpha)$) generate the $\mathbf{Z}$-module $E \otimes_A F$ (resp. $P$), $\pi' \circ \pi$ and $\pi \circ \pi'$ are the identity mappings.

Loosely speaking, Proposition 7 may be expressed by saying that tensor products commute with direct limits and usually the two sides of (7) are identified by means of the isomorphism $\pi$.