Lemma 2

Let $A$ be a ring, $\alpha$ an ideal of finite type of $A$, and $M$ an $A$-module each element of which is annihilated by a power of $\alpha$. Let $\widehat{A}$ denote the separated completion of $A$ for the $\alpha$-adic topology.

a) There exists on $M$ a unique structure of $\widehat{A}$-module extending the given structure of $A$-module.

b) *The sub-$\widehat{A}$-modules of $M$ are its sub-$A$-modules, and one has $\operatorname{Hom}_A(M, P) = \operatorname{Hom}_{\widehat{A}}(M, P)$ for every $\widehat{A}$-module $P$.

a) Let us identify $\widehat{A}$ with the projective limit of the rings $A/\alpha^n$, and endow $M$ with the discrete topology. Let $a = (a_n)$ be an element of $\widehat{A}$, and $x$ an element of $M$. Since $x$ is annihilated by a power of $\alpha$, the sequence $(a_n x)$ is stationary; let $ax$ denote its limit. The mapping $(a, x) \mapsto ax$ defines on $M$ a structure of $\widehat{A}$-module extending the given structure of $A$-module.

Conversely, suppose given such a structure on $M$; let $a = (a_n)$ be an element of $\widehat{A}$, $x$ an element of $M$ and $m$ an integer such that $\alpha^m x = 0$. For every integer $n$, $a - a_n$ belongs to $\alpha^n$, which is equal to $\alpha^n \widehat{A}$ (III, § 2, n° 12, cor. 2 of prop. 16); hence one has $ax = a_n x$ for $n \geq m$, whence the assertion of uniqueness.

b) It follows from the preceding that one has $Ax = \widehat{A}x$ for every $x \in M$; the $\widehat{A}$-submodules of $M$ are therefore its $A$-submodules. Finally, let $u$ be an $A$-linear homomorphism of $M$ into a $\widehat{A}$-module $P$. Let $a = (a_n)$ be an element of $\widehat{A}$, $x$ an element of $M$, and $m$ an integer such that $\alpha^m x = 0$; one has $\alpha^m u(x) = 0$. Since $a - a_m$ belongs to $\alpha^m \widehat{A}$, one has

so that $u$ is $\widehat{A}$-linear.