An example of almost etale extension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:34:38Z http://mathoverflow.net/feeds/question/114857 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114857/an-example-of-almost-etale-extension An example of almost etale extension kiseki 2012-11-29T08:05:53Z 2012-11-29T13:32:56Z <p>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:</p> <p>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.$V_{n}$).</p> <p>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.</p> <p>My question is:</p> <p>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 ? </p>