etale case new pt 1

article_de_rham_cyclic.tex

@@ -667,9 +667,20 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 \end{proof}
 
 \begin{proof}[Proof of Main Theorem]
-	As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k  \ol k$ by Lemma~\ref{lem:reductions}.
+	As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k = \ol k$ by Lemma~\ref{lem:reductions}.
 	We prove this by induction on $n$. If $n = 0$, then it follows by Chevalley--Weil theorem.
 	Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale.
+	Note that the proof of Lemma~\ref{lem:G_invariants_\'{e}tale} implies
+	that there exists an exact sequence:
+	%
+	\[
+		0 \to k \to H^1_{dR}(Y) \to T^1 \mc M \to k \to 0.
+	\]
+	%
+	Therefore, since the category of $k[C]$-modules is semisimple, $T^1 \mc M \cong H^1_{dR}(Y)$. By induction hypothesis we conclude that
+	$T^1 \mc M$ is determined by higher ramification data.
+	
+	
 	Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma - 1)$
 	is an isomorphism of $k$-vector spaces between $T^{i+1} \mc M$ and $T^i \mc M$ for
 	$i = 2, \ldots, p^n$. This yields an isomorphism of $k[C]$-modules for $i \ge 2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}: