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wrong pf of etale case

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jgarnek 2024-12-06 15:25:50 +01:00
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@ -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} \end{proof}
% %
\begin{Lemma} \label{lem:N+Nchi+...} \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 $i$. %If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$.
\end{Lemma} \end{Lemma}
\begin{proof} \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: By tensoring this isomorphism by $\chi^i$ we obtain:
% %
\begin{align*} \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}} 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 (N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \cong (M^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\chi^j}
\end{align*} \end{align*}
% %
for $i = 0, \ldots, p-2$. Therefore: for $i = 0, \ldots, p-2$. Therefore:
% %
\begin{equation} \label{eqn:N+M=M} \begin{equation} \label{eqn:M+N=N}
N^{\oplus p} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \oplus M^{\oplus p} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}
\cong M^{\oplus (p-1)}. \cong N^{\oplus (p-1)}.
\end{equation} \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*} \begin{align*}
N^{\oplus p} &\oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} M^{\oplus p} &\oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}
\cong N^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((N^{\chi^i})^{\oplus 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} N^{\chi^j} \right)\\ \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\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 \left( M^{\oplus 2} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \right)^{\oplus (p-1)}
\cong M^{\oplus (p-1)}. \cong N^{\oplus (p-1)}.
\end{align*} \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} \end{proof}
% %
\begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary} \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. 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. 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 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. $T^1 \mc M$ is determined by higher ramification data.
Recall now that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma - 1)$
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 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}: $i = 2, \ldots, p^n$. This yields an isomorphism of $k[C]$-modules for $i \ge 2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}:
% %
@ -718,13 +720,18 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
% %
\begin{align} \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^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} \end{align}
% %
Thus, since by induction hypothesis $\mc T^i \mc M$ is 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
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}, $T^1 \mc M$ \[
is also determined by higher ramification data. \mc T^1 \mc M/T^1 \mc M = H^1_{dR}(X)/H^1_{dR}(Y)
\]
%
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} 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: yield an isomorphism of $k[C]$-modules: