Finite étale algebras over a complete local ring are determined by reduction

By [F1] construct compatible finite étale algebras $D_n$ over $A\mathrm{/}{I}^n$. Each is free of the same finite rank $r$: its residue-field dimension is constant under reduction. Choose a basis of $D_1$ and lift it successively to $D_{n+1}$. Nakayama makes each lifted list a basis, since the source and target are free of the same rank and its determinant reduces to a unit. Thus these identifications respect the transition maps, and is $A^r$ as a module. The compatible multiplication tables and units define a commutative associative unital algebra structure on it. Its reduction modulo $I^n$ is $D_n$.

The algebra $D$ is finite free, hence flat and finitely presented as an algebra by [F1]. Its finite module of differentials has reduction modulo $I$ equal to zero by [F2], and Nakayama makes it zero. Thus $D$ is finite étale. Given two finite étale $A$-algebras and a map between their reductions, [F1] gives compatible maps modulo all $I^n$. Their matrices have a unique limit because finite free modules are complete and separated. The limit preserves multiplication and unit, since these identities hold modulo every $I^n$; conversely any map is determined by all those reductions. This proves the equivalence.

If $A$ is initially complete for its maximal ideal, it is complete for every ideal $I\subseteq \mathfrak{m}$. Choose representatives for a compatible system modulo $I^n$. They form a maximal-adic Cauchy sequence since $I^n\subseteq {\mathfrak{m}}^n$, and hence have a limit in $A$. Each $I^n$ is closed in the maximal-adic topology, because $A\mathrm{/}{I}^n$ is a complete separated finite module by [F3]; consequently the limit has every prescribed residue modulo $I^n$. Injectivity follows from $\bigcap I^n\subseteq \bigcap {\mathfrak{m}}^n=0$. This proves the parameter-ideal application in the Statement without an additional completeness assumption.