In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:
Let $k$ be a perfect field of characteristic $p$ and $W(k)$ the ring of Witt vectors, $W(k)(x_{1},...,x_{d})$ a localization of $W(k)[x_{1},...,x_{d}]$ at $p$. Let $V$ be the completion of $W(k)(x_{1},...,x_{2})$ with function field $K:=K(V)$, and $V_{n}$ be the extension of $V$ which generated by the $p^{n+1}$-th roots of unite together with $p^{n}$-th roots of $x_{i}$. If $W$ is a normalization of $V$ in another finite extension $L$ of $K$, and $W_{n}$ the composite of $W$ and $V_{n}$, and $p^{a_{n}}W_{n}$ the different of $W_{n}$ over $V_{n}$. If we use the notation $W_{\infty}$ (resp. $V_{\infty}$) to denote the union of $W_{n}$ (resp.$V_{n}$).
The Theorem 1.2 of Faltings' paper tell us $a_{n}$ converge to $0$ for $n \rightarrow \infty$, and then $V_{\infty} \longrightarrow W_{\infty}$ is almost etale extension.
My question is:
Why the almost etale extension of $W_{\infty}/V_{\infty}$ is deduced by theorem 1.2 ? Is this checked by the definition of almost etale ?
