irreducible -> indecomposable; ccorrect def of X''; Hasse Arf

article_de_rham_cyclic.tex

@@ -137,6 +137,9 @@ hyperref, bbm, mathtools, mathrsfs}
 	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}
+%
+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, see \cite{??Garnek_indecomposables}.
 
 \section{Cyclic covers}
 %
@@ -149,9 +152,11 @@ Let for any $\ZZ/p^n$-cover $X \to Y$
 %
 Note that if $G_P = \ZZ/p^n$, this coincides with the standard definition of
 the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$.
+If $G_P = \ZZ/p^m$, then (??relation with usual jumps??). By Hasse--Arf theorem (cf. ???),
+the numbers $u_{X/Y, P}^{(t)}$ are integers.
 %
 \begin{Theorem} \label{thm:cyclic_de_rham}
-	Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m = G_{P_0}$ for $P_0 \in X(k)$. Then, as $k[\ZZ/p^n]$-modules:
+	Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m$. Pick arbitrary $P_0 \in X(k)$ such that $G_{P_0} \cong \ZZ/p^m$. Then, as $k[\ZZ/p^n]$-modules:
 	%
 	\[
 	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} J_{p^n - p^n/e_P}^2
@@ -171,7 +176,7 @@ For any $k[H]$-module $M$ denote:
 %
 Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????).
 In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case
-we denote the irreducible $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
+we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
 and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\
 Note also that for $j \ge 1$:
 %
@@ -260,8 +265,8 @@ We define $m_{\sigma - 1}$ as follows:
 	m_{\sigma - 1}(\ol x) := (\sigma - 1) \cdot x,
 \]
 %
-where for $\ol x \in T^i M$ we picked any representative $x \in M^{(i)}$.
-Indeed, if $x \in M^{(i+1)} := \ker((\sigma - 1)^{i+1})$ then clearly $(\sigma - 1) x \in M^{(i)}$.
+where for $\ol x \in T^{i+1} M$ we picked any representative $x \in M^{(i+1)}$.
+Indeed, if $x \in M^{(i+1)}$ then clearly $(\sigma - 1) \cdot x \in M^{(i)}$.
 Moreover $(\sigma - 1) \cdot x \in M^{(i-1)}$ holds if and only if $x \in M^{(i)}$. This
 shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
@@ -275,7 +280,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	\]
 \end{Lemma}
 \begin{proof}
-	By Lemma~\ref{lem:TiM_isomorphism}:
+	Note that $\mc T^i M = M^{(pi)}/M^{(pi - p)}$. This easily implies that:
 	%
 	\begin{align*}
 		\dim_k \mc T^i M &= \dim_k T^{pi} M + \ldots + \dim_k T^{pi - p + 1} M\\
@@ -288,7 +293,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 
 \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
 	We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
-	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$.
+	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$.
 	Write also $\mc M := H^1_{dR}(X)$.
 	We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$:
 	%
@@ -329,12 +334,13 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 		\oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
 	\]
 	%
-	where $e'_P := e_{X/Y', P}$. Note that
+	where $e'_P := e_{X/Y', P}$. Note that for any $P \in X(k)$:
 	%
 	\[
-		??? p \cdot u^{(n)} = u^{(n-1)} + (p-1) \cdot l^{(1)} ???.
+		p \cdot u^{(n)}_{X/Y, P} = u^{(n-1)}_{X/Y', P} + (p-1) \cdot l^{(1)}_{Y'/Y, Q},
 	\]
 	%
+	where $Q$ denotes the image of~$P$ in~$Y'$.
 	Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. ????):
 	%
 	\begin{align*}
@@ -342,7 +348,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 		2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\
 		&= 2 p (g_Y - 1) + \sum_{Q \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q}^{(1)} + 1)\\
 		 &+ 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\
-		&= ?? p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1) \right)
+		&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1) \right)
 	\end{align*}
 	%
 	where