wrong pf of etale case

article_de_rham_cyclic.tex

@@ -614,47 +614,51 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 \end{proof}
 %
 \begin{Lemma} \label{lem:N+Nchi+...}
-	Keep the above notation. Let $M$, $N$ be $k[C]$-modules. Assume that
+	Keep the above notation. Let $M$, $N$ be $k[C]$-modules. 
+	\begin{enumerate}[(1)]
+		\item If $N \cong M \oplus M^{\chi} \oplus \ldots \oplus M^{\chi^{p-1}},$
+		then $N$ is uniquely determined by $M$.
+		
+		\item If $N^{\oplus (p-1)} \cong M \oplus M^{\chi} \oplus \ldots \oplus M^{\chi^{p-2}}$,
+		then $M \cong N \cong N^{\chi}$.
+	\end{enumerate}
 	%
-	\[
-		M \cong N \oplus N^{\chi} \oplus \ldots \oplus N^{\chi^{p-1}}.
-	\]
-	%	
-	Then $N$ is uniquely determined by $M$.
+	
 	%If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$.
 \end{Lemma}
 \begin{proof}
-	Note that
+	(1) Note that
 	%
 	\[
-		M \cong N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}.
+	N \cong M^{\oplus 2} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}}.
 	\]
 	%
 	By tensoring this isomorphism by $\chi^i$ we obtain:
 	%
 	\begin{align*}
-		M^{\chi^i} \cong (N^{\chi^i})^{\oplus 2} \oplus N^{\chi^{i+1}} \oplus N^{\chi^{i+2}} \oplus \ldots \oplus N^{\chi^{i + p-2}}
-		\cong (N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j}
+		N^{\chi^i} \cong (M^{\chi^i})^{\oplus 2} \oplus M^{\chi^{i+1}} \oplus M^{\chi^{i+2}} \oplus \ldots \oplus M^{\chi^{i + p-2}}
+		\cong (M^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\chi^j}
 	\end{align*}
 	%
 	for $i = 0, \ldots, p-2$. Therefore:
 	%
-	\begin{equation} \label{eqn:N+M=M}
-		N^{\oplus p} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \oplus 
-		\cong M^{\oplus (p-1)}.
+	\begin{equation} \label{eqn:M+N=N}
+		M^{\oplus p} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} 
+		\cong N^{\oplus (p-1)}.
 	\end{equation}
 	%
-	Indeed, for the proof of~\eqref{eqn:N+M=M} note that
+	Indeed, for the proof of~\eqref{eqn:M+N=N} note that
 	%
 	\begin{align*}
-		N^{\oplus p} &\oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}}
-		\cong N^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((N^{\chi^i})^{\oplus 2}
-		\oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \right)\\
-		&\cong \left( N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} \right)^{\oplus (p-1)}
-		\cong M^{\oplus (p-1)}.
+		M^{\oplus p} &\oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}
+		\cong M^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((M^{\chi^i})^{\oplus 2}
+		\oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\chi^j} \right)\\
+		&\cong \left( M^{\oplus 2} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \right)^{\oplus (p-1)}
+		\cong N^{\oplus (p-1)}.
 	\end{align*}
 	%
-	The isomorphism~\eqref{eqn:N+M=M} clearly proves the thesis.
+	The isomorphism~\eqref{eqn:M+N=N} clearly proves the thesis.\\
+	(2) 
 \end{proof}
 %
 \begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary}
@@ -695,17 +699,15 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	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:
+	that there exists an exact sequence of $k[C]$-modules:
 	%
 	\[
-		0 \to k \to H^1_{dR}(Y) \to T^1 \mc M \to k \to 0.
+		0 \to k \to H^1_{dR}(Y) \to T^1 \mc M \to k \to 0,
 	\]
 	%
-	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
+	where $k$ denotes the trivial $k[C]$-module. 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 $(\sigma - 1)$
+	Recall now 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}:
 	%
@@ -718,14 +720,19 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	%
 	\begin{align}
 		\mc T^1 \mc M &\cong T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}} \label{eqn:decomposition_of_mc_T1}\\
-		\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^{n-1}. ???? \label{eqn:decomposition_of_mc_Ti}
+		\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^{n-1}. \label{eqn:decomposition_of_mc_Ti}
 	\end{align}
 	%
-	Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by higher ramification data,
-	we have by Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} that $T^2 \mc M$ is determined by higher ramification data.
-	Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$
-	is also determined by higher ramification data.
-
+	If $n \ge 2$, then by induction hypothesis, Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} the $k[C]$-module $T^2 \mc M$ is determined by higher ramification data. If $n = 1$, then the $k[C]$-module
+	%
+	\[
+		\mc T^1 \mc M/T^1 \mc M = H^1_{dR}(X)/H^1_{dR}(Y)
+	\]
+	%
+	is of the form $N^{\oplus (p-1)}$ for a $k[C]$-module $N$
+	by Lemma~????. Therefore by Lemma~????? we have $T^2 \mc M \cong N$.
+	This ends the proof, as $N$ is determined by the fundamental characters of the group action of $C$ on $X$.\\
+	
 	Assume now that $X \to Y$ is not \'{e}tale. Analogously as in the previous case, Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham}
 	yield an isomorphism of $k[C]$-modules:
 	%