Finite etale schemes over a complete local ring and splitting

Write $Y=\mathrm{Spec}A$ with $A$ a finite etale $R$-algebra. By [F1] the reduction functor $A\mapsto A{\otimes }_{R}k=A\mathrm{/}\mathfrak{m}A$ is an equivalence from finite etale $R$-algebras to finite etale $k$-algebras. An equivalence is fully faithful, so it induces bijections ${\mathrm{Hom}}_{R\text{-alg}}(A,R)⟶{\mathrm{Hom}}_{k\text{-alg}}(A{\otimes }_{R}k,k),$ natural in $A$; under the anti-equivalence of affine schemes these are the maps $Y(R)\to Y(k)$ given by reduction. Hence $Y(R)\to Y(k)$ is a bijection.

Assume now that $k$ has no nontrivial finite separable extension, and let $E$ be a finite etale $k$-algebra. As a finite-dimensional commutative $k$-algebra, $E$ is Artinian, so $E\cong {\prod }_i{E}_i$ with each $E_i$ local Artinian by [F4]. Each $E_i$ is a direct factor of $E$, hence finite etale over $k$; being etale over the field $k$ it is smooth of relative dimension zero, so its only fibre is geometrically regular and in particular regular by [F5]. A regular local ring is a domain, and a local Artinian domain is a field by [F4], so $E_i$ is a field; as a finite etale field extension of $k$ it is finite separable over $k$ by [F3], hence equals $k$. Therefore $E\cong k^d$ for $d={\dim }_{k}E$, and every finite etale $k$-algebra is a product of copies of $k$.

Let $A$ be a finite etale $R$-algebra of rank $d$; by [F2] its rank equals the $k$-dimension of $A{\otimes }_{R}k$, which is a finite etale $k$-algebra, so $A{\otimes }_{R}k\cong k^d$ by step 1.2. Since reduction is an equivalence by [F1], it is essentially surjective and reflects isomorphisms, so $A\cong R^d$; consequently $Y=\mathrm{Spec}A$ is the disjoint union of $d$ copies of $\mathrm{Spec}R$. The complete Noetherian local hypotheses are exactly those stated, and AC is available for both the lifting equivalence [F1] and the classification suppliers [F2]-[F4].