Corollary

Under the hypotheses of Proposition 9, let M be an A-module. We write $f_{\beta \alpha} = i^{S_\beta, S_\alpha}_{M}$ for $\alpha \leq \beta, f_\alpha = i^{S, S_\alpha}_{M}$ for all $\alpha \in I$ and let $f'_\alpha$ be the canonical mapping from $S_\alpha^{-1}M$ to $\lim \rightarrow S_\alpha^{-1}M$; then there exists an $S^{-1}A$-isomorphism $g$ of $S^{-1}M$ onto $\lim \rightarrow S_\alpha^{-1}M$ such that $g \circ f_\alpha = f'_\alpha$ for all $a \in I$.

The corollary follows immediately from the definitions $S_\alpha^{-1}M = M \otimes_A S_\alpha^{-1}A$ and $S^{-1}M = M \otimes_A S^{-1}A$ and the fact that taking direct limits commutes with tensor products (Algebra, Chapter 11, § 6, no. 3, Proposition 7).