sketch of pf for Zpn

article_de_rham_cyclic.out

@@ -1,3 +1,4 @@
-\BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040}{}% 1
+\BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1
 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 2
-\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 3
+\BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000H\000y\000p\000o\000e\000l\000e\000m\000e\000n\000t\000a\000r\000y\000\040\000c\000o\000v\000e\000r\000s}{}% 3
+\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 4

article_de_rham_cyclic.tex

@@ -112,7 +112,7 @@ hyperref, bbm, mathtools, mathrsfs}
 %opening
 \begin{document}
 
-\title[The de Rham...]{?? The de Rham cohomology of covers with cyclic $p$-Sylow subgroup}
+\title[The de Rham...]{?? The de Rham cohomology of covers\\ with cyclic $p$-Sylow subgroup}
 \author[A. Kontogeorgis and J. Garnek]{Aristides Kontogeorgis and J\k{e}drzej Garnek}
 \address{???}
 \email{jgarnek@amu.edu.pl}
@@ -129,8 +129,15 @@ hyperref, bbm, mathtools, mathrsfs}
 \maketitle
 \bibliographystyle{plain}
 %
-\section{}
+\section{Introduction}
 %
+\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 closed
+	points $x$ of $X$ that are ramified in the cover $X \to X/G$.
+\end{mainthm}
+
 \section{Cyclic covers}
 %
 Let for any $\ZZ/p^n$-cover $X \to Y$
@@ -206,7 +213,17 @@ Note also that for $j \ge 1$:
 	????
 \end{proof}
 %
-
+\begin{Lemma}
+	For any $i \le p^n - 1$:
+	%
+	\[
+		(\sigma - 1) : T^{i+1} M \hookrightarrow T^i M.
+	\]
+\end{Lemma}
+\begin{proof}
+	
+\end{proof}
+%
 \begin{proof}[Proof of Theorem ????]
 	We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
 	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$.
@@ -224,7 +241,7 @@ Note also that for $j \ge 1$:
 	m' :=
 		\begin{cases}
 			n-1, & \textrm{ if } m = n,\\
-			n, & \textrm{ otherwise.}
+			m, & \textrm{ otherwise.}
 		\end{cases}
 	\]
 	
@@ -236,7 +253,50 @@ Note also that for $j \ge 1$:
 			???, 
 		\end{cases}
 	\end{align*}
+	%
+	In particular, $\dim_k \mc T^1 M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} M$.
+	On the other hand, by Lemma ??:
+	%
+	\begin{align*}
+		\dim_k \mc T^1 M &= \dim_k T^1 M + \ldots + \dim_k T^p M\\
+		&\ge \dim_k T^{p^n - p^{n-1}} M + \ldots + \dim_k T^{p^n - p^{n-1}} M
+		= \dim_k \mc T^{p^{n-1} - p^{n-2}} M.
+	\end{align*}
+	%
+	Since the left-hand side and right hand side are equal, we conclude by Lemma ???
+	that
+	%
+	\[
+		\dim_k T^1 M = \ldots = \dim_k T^{p^n - p^{n-1}} M = \frac{1}{p} \dim_k \mc T^1 M.
+	\]
+	%
+	If the cover $X \to X''$ is \'{e}tale, then the cover $X \to Y$ must be also \'{e}tale.
+	Thus the proof follows in this case by~\cite{Nakajima??Inventiones}. Suppose now that 
+	$X \to X''$ is not \'{e}tale. Then, by Lemma ???, the map $\tr_{X/X''} : H^1_{dR}(X) \to H^1_{dR}(X'')$ is surjective. Moreover, note that in the group ring $k[H]$ we have:
+	%
+	\[
+		\tr_{X/X''} = \sum_{j = 0}^{p-1} (\sigma^{p^{n-1}})^j = (\sigma^{p^{n-1}} - 1)^{p-1} = 
+		(\sigma - 1)^{p^n - p^{n-1}}.
+	\]
+	%
+	This implies that:
+	%
+	\[
+		\ker(\tr_{X/X''} : M \to M'') = M^{(p^n - p^{n-1})}
+	\]
+	%
+	and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} M \to \mc T^i M''$ for any $i \ge 1$. Thus:
+	%
+	\[
+		\dim_k T^{i + p^n - p^{n-1}} M = \dim_k \mc T^i M'' = ....
+	\]
+	%
+	This ends the proof.
 \end{proof}
 
+\section{Hypoelementary covers}
+%
+Assume now that $G = H \rtimes_{\chi} \ZZ/??n$.
+
 \bibliography{bibliografia}
 \end{document}