intro - higher ramification data

article_de_rham_cyclic.tex

@@ -142,17 +142,17 @@ Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set o
 This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of indecomposable modules is finite (cf. \cite{Higman}, \cite{Borevic_Faddeev}, \cite{Heller_Reiner_Reps_in_integers_I}). While their representation theory
 seems too complicated to derive a formula for the cohomologies, 
 the article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} proved that 
-the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the lower ramification groups and the fundamental characters of the ramification locus. The main result of this article is a similar statement for the de Rham cohomology.
+the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data (i.e. higher ramification groups and the fundamental characters of the ramification locus). The main result of this article is a similar statement for the de Rham cohomology.
 %
 \begin{mainthm}
 	Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
 	Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
-	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the lower ramification groups and the fundamental characters of
+	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the
-	points $P \in X(k)$ that are ramified in the cover $X \to X/G$.
+	higher ramification data of the cover $X \to X/G$.
 \end{mainthm}
 %
 Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
-of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data. Indeed, see \cite{garnek_indecomposables} for a construction of $G$-covers with the same ramification data, but different $k[G]$-module structure of $H^0(X, \Omega_X)$ and
+of $H^1_{dR}(X)$ isn't determined uniquely by the higher ramification data. Indeed, see \cite{garnek_indecomposables} for a construction of $G$-covers with the same higher ramification data, but different $k[G]$-module structure of $H^0(X, \Omega_X)$ and
 $H^1_{dR}(X)$.
 
 \section{Cyclic covers}
@@ -466,7 +466,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	and note that $\dim_k T^i J_l = \llbracket i \le l \rrbracket$ for $i > 0$. Therefore, if $i \in (p^n - p^{N+1}, p^n - p^N]$:
 	%
 	\begin{align*}
-		\dim_k T^i \mc M &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
+		\dim_k T^i \mc M_0 &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
 		&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\
 		&+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_Q - 1} \llbracket t \ge n_Q + N - n \rrbracket \cdot (u_{Q}^{(t+1)} - u_{Q}^{(t)}).
 	\end{align*}
@@ -615,10 +615,10 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 		\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 - p^{n-1}. \label{eqn:decomposition_of_mc_Ti}
 	\end{align}
 	%
-	Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by ramification data,
+	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 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 ramification data.
+	is also determined by higher ramification data.
 
 	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:
@@ -636,7 +636,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 		(T^1 \mc M)^{\chi^{-p}}.
 	\end{align*}
 	%
-	By induction assumption, the $k[C]$-module structure of $\mc T^i \mc M$ is uniquely determined by the ramification data. Thus, by Lemma~\ref{lem:N+Nchi+...} for $N := T^1 \mc M$ and by~\eqref{eqn:TiM=T1M_chi} the $k[C]$-structure of the modules $T^i \mc M$ is uniquely determined by the ramification data for $i \le p^n - p^{n-1}$.
+	By induction assumption, the $k[C]$-module structure of $\mc T^i \mc M$ is uniquely determined by the higher ramification data. Thus, by Lemma~\ref{lem:N+Nchi+...} for $N := T^1 \mc M$ and by~\eqref{eqn:TiM=T1M_chi} the $k[C]$-structure of the modules $T^i \mc M$ is uniquely determined by the higher ramification data for $i \le p^n - p^{n-1}$.
 	By a similar reasoning, $\tr_{X/X'}$ yields an isomorphism:
 	%
 	\[
@@ -644,7 +644,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	\]
 	%
 	Thus, by induction hypothesis for $\mc M''$, the $k[C]$-structure of $T^{i + p^n - p^{n-1}} \mc M$
-	is determined by ramification data as well.
+	is determined by higher ramification data as well.
 \end{proof}
 
 \section{Proof of Main Theorem}