Exercise 1, § 4

Let A be a commutative ring which is Hausdorff and complete with a filtration $(\mathfrak{a}_n)_{\geq 0}$ such that $\mathfrak{a}_0 = A$. Let M, M', N be three A-modules, each with the filtration induced by that on A and the topology defined by that filtration; we write $\overline{M} = M/\mathfrak{a}_1 M, \overline{M}' = M'/\mathfrak{a}_1 M', \overline{N} = N/\mathfrak{a}_1 N$. Let $f : M \times M' \to N$ be a bilinear mapping and $\tilde{f} : \overline{M} \times \overline{M}' \to \overline{N}$ the $(A/\mathfrak{a}_1)$-bilinear mapping derived from $f$ by taking quotients. Let $y \in N, \bar{x} \in \overline{M}, \bar{x}' \in \overline{M}'$ be such that: (1) $\tilde{f}(\bar{x}, \bar{x}')$ is the class $\bar{y}$ of $y$ in $\overline{N}$; (2) every element of $\overline{N}$ can be written as

where $\bar{z} \in \overline{M}$ and $\bar{z}' \in \overline{M}'$. Show that, if N is Hausdorff and M and M' are complete, there exists $x \in \bar{x}$ and $x' \in \bar{x}'$ such that $f(x, x') = y$ (argue by induction as in the proof of Hensel's lemma). Under what conditions are $x$ and $x'$ determined uniquely?