I'm having trouble understanding the proof of Proposition 3.2.5 (i) of "Etale Cohomology Theory" by Lei Fu. The statement of this proposition is as follows:
Proposition 3.2.5. Let $X$ be a scheme on which a finite group $G$ acts admissibly on the right and let $Y=X/G$ be the quotient of $X$ by $G$. Suppose $X$ is of finite presentation over $Y$. If the inertia group $G_i(x)$ is trivial for any $x\in X$, then $X$ is etale over $Y$.
In the proof of this, the author first takes the strict henselization of $\mathcal{O}_{Y,y}$ with respect to a separable closure of $k(y)$ for arbitrary $y\in Y$, which is denoted by $\widetilde{\mathcal{O}}_{Y,\bar{y}}$, and then he mentions Proposition 2.5.10,whose statement is as follows:
Proposition 2.5.10. Consider a Cartesian diagram
$\require{AMScd}$
\begin{CD}
X'=X\times_Y Y' @>{g'}>> X\\
@V{f'}VV @V{f}VV\\
Y' @>{g}>> Y
\end{CD}
Suppose that $f:X\to Y$ is locally of finite presentation and $g:Y'\to Y$ is faithfully flat. If $f'$ is unramified (resp. etale, resp. smooth), then so is $f$.
Using this, he sais that it suffices to show that $X\times_Y\mathrm{Spec}\,\widetilde{\mathcal{O}}_{Y,\bar{y}}$ is etale over $\mathrm{Spec}\,\widetilde{\mathcal{O}}_{Y,\bar{y}}$ and reduces to the case where $Y$ is a spectrum of a strictly henselian local ring. What I don't understand is this point. Generaly, the map $\mathrm{Spec}\,\widetilde{\mathcal{O}}_{Y,\bar{y}}\to Y$ is not faithfully flat, so I tried to reduce to the case in where $Y$ is a local scheme by replacing $X$ by $\coprod_{g\in G} \mathcal{O}_{X,x^g}$(on which $G$ acts on the right) and $Y$ by $\mathrm{Spec}\,\mathcal{O}_{Y,y}$. However I can't prove that the quotient of $\coprod_{g\in G} \mathcal{O}_{X,x^g}$ by $G$ is equal to $\mathrm{Spec}\,\mathcal{O}_{Y,y}$ (or perhaps is this a wrong statement?). If this statement is wrong, what is the right approach to proving this proposition?
Sorry for my poor English, and I really appreciate your help.
Best Answer
A morphism $f:X \to Y$ is étale if and only if it for all $y \in Y$ the basechange of $f$ along $\operatorname{Spec} \mathcal{O}_{Y,y}$ is étale. Lets take this for granted (c.f. https://stacks.math.columbia.edu/tag/02GU); the proof depends on your definition of étale. Your proposition 2.5.10 tells us that we can check étaleness after further basechange to $\overline{\operatorname{Spec}} \mathcal{O}_{Y,\overline{y}}$.