Let $E$ be a non-archimedean field, $\mathcal O$ its ring of integers, $\pi \in \mathcal O$ a uniformizer so that $\pi\mathcal O$ is the maximal ideal of $\mathcal O$, and let $\kappa = \mathcal O/\pi\mathcal O$ be the residue field. For a formal scheme $\mathfrak X$ over $\mathcal O$ and for $n\geq 1$, let $\mathfrak X_n$ denote the scheme $(\mathfrak X, \mathcal O_{\mathfrak X}/\pi^n \mathcal O_{\mathfrak X})$ ; it is a scheme over $\mathcal O / \pi^n \mathcal O$. In particuler, $\mathfrak X_1 = \mathfrak X_s$ is the special fiber of $\mathfrak X$.
In Berkovich's paper "Vanishing cycles for formal schemes" , a morphism $\varphi: \mathfrak Y \to \mathfrak X$ of formal schemes over $\mathcal O$ is said to be etale if for every $n\geq 1$, the induced morphism of schemes $\varphi_n: \mathfrak Y_n \to \mathfrak X_n$ is étale.
In the appendix of the book "Motivic integration" by Chambert-Loir, Nicaise and Sebag, a morphism $\varphi: \mathfrak Y \to \mathfrak X$ is said to be etale if it is formally etale and locally formally of finite type. The property of being formally etale is defined using the functor of points associated to formal schemes.
I take it that both definitions should be equivalent, and it sounds reasonable given that similar definitions in the case of schemes are equivalent indeed.
However, here is something puzzling me.
After introducing the definition of etale morphisms in the book, the authors give the following example.
If $\mathfrak X$ is a locally noetherian formal scheme and $Z$ is a subscheme of $\mathfrak X_{\mathrm{red}} := (\mathfrak X_s)_{\mathrm{red}}$, then the completion morphism
$$\varphi_Z:\widehat{\mathfrak X\setminus Z} \to \mathfrak X$$
is etale.
The formal scheme $\widehat{\mathfrak X\setminus Z}$ denotes the formal completion of $\mathfrak X$ along $Z$. It is a formal scheme whose special fiber is $Z$.
If the morphism $\varphi_Z$ is etale, then according to Berkovich's definition the morphism of schemes $(\varphi_Z)_s: Z \to \mathfrak X_s$, which factors through a morphism $Z \to \mathfrak X_{\mathrm{red}}$, is etale. But etale morphisms are open ; so shouldn't this imply that $Z$ is an open subscheme of $\mathfrak X_{\mathrm{red}}$ ? It feels like something is wrong with this example…
Best Answer
I believe the example you quote from Motivic Integration is correct. For example, the map $\mathrm{Spf}(\kappa[[x]]) \rightarrow \mathrm{Spec}(\kappa[x])$ (formal completion of the affine line at the closed point $x=0$) is both locally formally of finite type and formally étale.
Regarding comparison with the definition you quote from Berkovich (I didn't look at the mentioned paper), the issue is that when you say that $\mathfrak{X}_n$ are schemes, you are asssuming that $\mathfrak{X}_n \rightarrow \mathrm{Spf}(\mathcal{O})$ is an adic morphism (say, of locally Noetherian formal schemes). This also forces your $\varphi \colon \mathfrak{Y} \rightarrow \mathfrak{X}$ to be an adic morphism (again, say of locally Noetherian formal schemes). The definition you quote from Motivic Integration treats not necessarily adic morphisms (like $\mathrm{Spf}(\kappa[[x]]) \rightarrow \mathrm{Spec}(\kappa[x])$ from above).
For an adic morphism of locally Noetherian formal schemes, the two definitions will agree.