Remark

(1) Proposition 1 and its proof are valid for arbitrary groups, except for change of notation.

(2) Note that if there are exact sequences

it does not necessarily follow that the sequence

is exact; in other words, the inverse limit of an inverse system of surjective linear mappings is not necessarily surjective (cf. Exercise 1).

Suppose now that the $A_\alpha$ are equal to the same ring $A$ and the $\phi_{\alpha\beta}$ to $1_A$; then for every inverse system $(E_\alpha, f_{\alpha\beta})$ of $A$-modules, $E = \lim E_\alpha$ is an $A$-module. Let $F$ be an $A$-module and, for all $\alpha$, let $u_\alpha : F \to E_\alpha$ be an $A$-linear mapping such that $(u_\alpha)$ is an inverse system of mappings; then $u = \lim u_\alpha$ is an $A$-linear mapping of $F$ into $\lim E_\alpha$. Conversely, for every $A$-linear mapping $v : F \to \lim E_\alpha$, the family of $v_\alpha = f_\alpha \circ v$ is an inverse system of $A$-linear mappings such that $v = \lim_{\leftarrow} v_\alpha$. We note on the other hand that for $\alpha \leq \beta$ the mapping

is a $\mathbf{Z}$-module homomorphism such that $(\operatorname{Hom}_A(F, E_\alpha), \bar{f}_{\alpha \beta})$ is an inverse system of $\mathbf{Z}$-modules; as $\bar{f}_{\alpha \beta}(v_\beta) = f_{\alpha \beta} \circ v_\beta$, the above remarks can therefore be expressed as follows: