Let $X:=\operatorname{Spec}(R)$ an affine Noetherian scheme and $G$ a finite group acting on $X$. Then it is known that the quotient $Y=X/G$ exists as affine scheme $\operatorname{Spec}(R^G)$, let set $A:=R^G$ and denote the canonical projection map $p: X \to Y$. This map is finite.
Let us fix a point $x \in X$ and $y=p(x) \in Y$ and let introduce the decomposition group of $x$ by $D:=D(x)= \{g \in G \ | \ g(x)= x \}$.
Let consider the stalk map $s_x: \mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$. The group $D$ acts on $\mathcal{O}_{X,x}$, as well as on $\widehat{\mathcal{O}}_{X,x}$, the completion of the local ring wrt it's maximal ideal.
One can show that $(\widehat{\mathcal{O}}_{X,x})^D \cong \widehat{\mathcal{O}}_{Y,y}$ (analyze $G$-invariants $((R \otimes_{R^G} \mathcal{O}_{Y,y}) \otimes \widehat{\mathcal{O}}_{Y,y})^G$ und note that by finiteness of $R$ over $R^G$ we have $(R \otimes_{R^G} \mathcal{O}_{Y,y}) \otimes \widehat{\mathcal{O}}_{Y,y} = \bigoplus_{z \in p^{-1}} \widehat{\mathcal{O}}_{X,z}$
Especially this implies that $ \mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}^D$ is etale.
Now the question: In Qing Liu's book 'Algebraic Geometry' is claimed (Exercise 4.3.18 (b)) that if we moreover assume $G= D$, then it even holds $\mathcal{O}_{X,x}^D = \mathcal{O}_{Y,y}$ and I not understand why the additional assumption $G= D$ is neccessary.
Indeed, $\mathcal{O}_{X,x}^D \otimes_{\mathcal{O}_{Y,y}} \widehat{\mathcal{O}}_{Y,y} = (\mathcal{O}_{X,x} \otimes_{\mathcal{O}_{Y,y}} \widehat{\mathcal{O}}_{Y,y})^D= (\widehat{\mathcal{O}}_{X,x})^D= \widehat{\mathcal{O}}_{Y,y}= \mathcal{O}_{Y,y} \otimes \widehat{\mathcal{O}}_{Y,y}$ (first eq follows from flatness of $\widehat{\mathcal{O}}_{Y,y}$ over $\mathcal{O}_{Y,y}$, second from finiteness of $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$, third we discussed above, last ähm ... trivial.
Now since $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}^D$ is finite, too and $\widehat{\mathcal{O}}_{Y,y}$ over $\mathcal{O}_{Y,y}$ even faithfully flat, we can conclude reversly that $\mathcal{O}_{X,x}^D = \mathcal{O}_{Y,y}$ holds without assuming $G=D$.
What did I wrong in my reasonings above? Seemingly I nowhere used this additional assumption, so I conjecture that I have overseen somewhere a subtlety.
By the way does somebody know a good source discussing such kind of analysis on action by abstract finite group on local rings? Are there any generalizations to profinite groups (eg Galois groups of $Frac(R)$) acting on affine schemes known, with posed additional assumption that the quotient still exists, where it is known that $(\widehat{\mathcal{O}}_{X,x})^D \cong \widehat{\mathcal{O}}_{Y,y}$ still holds? in last sentence I have this kind of problems in mind.
