Lifting points of étale group scheme.

Question

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the completion of an algebraic extension, even infinite, of $K$, and let $S$ be the integral closure of $R$ inside $L$. What I want to prove is that if I take an étale finite group scheme $G$, represented by the étale Hopf algebra $A$, I have a bijection of sets \begin{equation} G(S/\mathfrak{m}^{i+1}S):=\text{Hom}_{\text{R-Alg}}(A,S/\mathfrak{m}^{i+1}S)\simeq \text{Hom}_{\text{R-Alg}}(A,S/\mathfrak{m}^{i}S):=G(S/\mathfrak{m}^{i}S) \end{equation} given for every $i$ by the composition with the projection to the next quotient. I guess that the main ingredient to prove this could be Hensel Lemma, since $R$ is Henselian, but I really don't see the proof. Any hint will be very appreciated. Thanks!

Answer 1

What you want is sometimes the definition of an étale morphism. You have a map $\phi: S/\mathfrak{m}^{i+1}S \rightarrow S/\mathfrak{m}^{i}S$ whose kernel $I=\mathfrak{m}^i/\mathfrak{m}^{i+1}$ satisfies $I^2=0$, and you're trying to show $$ \phi_*: \mathrm{Hom}_{R-Alg}(A,S/\mathfrak{m}^{i+1}S) \rightarrow \mathrm{Hom}_{R-Alg}(A,S/\mathfrak{m}^{i}S)$$

is bijective. This is called being formally étale. An étale ring homomorphism can be defined as one that is formally étale and essentially of finite presentation. See this question, for example.

The point is that ring maps with kernels $I$ satisfying $I^2=0$ correspond to little deformations. An étale morphism is supposed to be an analogue of a map of complex analytic varieties that is a local isomorphism. One way to interpret that is if you are given a point in the domain of a local isomorphism, there is a bijection between a small neighbourhood of that point and its image. Algebraically formulated, this becomes a formally étale morphism,