chevalley weil for de Rham; proof

article_de_rham_cyclic.tex

@@ -155,8 +155,7 @@ the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ra
 \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
-	higher ramification data of the cover $X \to X/G$ and the genus of $X$.
+	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the {\color{red} higher ramification groups} of the cover $X \to X/G$ and the genus of $X$.
 \end{mainthm}
 %
 Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
@@ -187,10 +186,7 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
 	\item $u^{(0)}_{X/Y, P} := 1$ for any ramified point $P \in X(\ol k)$
 	(note that this is not a standard convention),
 	
-	\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump,
-	
-	\item $\theta_{X/Y, P} : G_P \to \Aut_k(\mf m_P/\mf m_P^2) \cong k^{\times}$
-	is the fundamental character of~$P$.
+	\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump.
 \end{itemize}
 %
 By Hasse--Arf theorem (cf. 
@@ -231,32 +227,75 @@ $k[C]$-module. It turns out that the map
 is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write
 $\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
 
-Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, denote by $\chi_e$ the primitive character of a cyclic group of order $e$.
+Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, denote by $\chi_e$ the primitive character of a cyclic group of order $e$. Let also $\theta_{X/Y, P} : G_P \to \Aut_k(\mf m_P/\mf m_P^2) \cong k^{\times}$ be the fundamental character of~$P$. Again, for $Q \in Y(k)$ we write $\theta_{X/Y, Q} := \theta_{X/Y, P}$ for any $P \in \pi^{-1}(Q)$.
 %
 {\color{red}
 \begin{Proposition} \label{prop:chevalley_weil}
 	Keep the above notation and assume that $p \nmid \# G$. Then:
 	%
 	\begin{equation} \label{eqn:cw}
-		H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a_W},
+		H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a(X, G, W)},
 	\end{equation}
 	%
 	where:
 	%
 	\begin{align*}
-		a_W := (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{Q, i}(W) + \llbracket W \cong k \rrbracket,
+		a(X, G, W) := (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{Q, i}(W) + \llbracket W \cong k \rrbracket,
 	\end{align*}
 	%
 	and $N_{Q, i}(W)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $W \otimes_{k[G_Q]} \theta_{X/Y, Q}$.
 \end{Proposition}
 %
-\begin{Corollary}
+\begin{Corollary}[Chevalley--Weil formula for the de Rham cohomology]
 	Keep the notation of Proposition~\ref{prop:chevalley_weil}. Then:
 	\begin{equation} \label{eqn:cw_dR}
-		H^1_{dR}(X) \cong k[G]^{\oplus 2g_X - 2} \oplus k^{\oplus 2}.
+		H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a^{dR}(X, G, W)}.
 	\end{equation}
 	%
+	where:
+	%
+	\begin{align*}
+		a^{dR}(X, G, W) := 2 (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \dim_k W/W^{G_Q} + 2 \cdot \llbracket W \cong k \rrbracket.
+	\end{align*}
+	%
 \end{Corollary}
+\begin{proof}
+	Note that the category of $k[C]$-modules is semisimple. Hence, by the Hodge--de Rham exact sequence (??recall it earlier??) and Serre's duality (cf. ????):
+	%
+	\begin{align*}
+		H^1_{dR}(X) &\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)\\
+		&\cong H^0(X, \Omega_X) \oplus H^0(X, \Omega_X)^{\vee}\\
+		&\cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus (a(X, G, W) + a(X, G, W^{\vee}))}.
+	\end{align*}
+	%
+	Note moreover that $N_{Q, i}(W^{\vee}) = N_{Q, e_Q - i}(W)$
+	(since $\chi_{e_Q}^{e_Q - i}$ is the dual representation to $\chi_{e_Q}^i$) and:
+	%
+	\[
+		\sum_{i = 0}^{e_Q - 1} N_{Q, i}(W) = \dim_k W.
+	\]
+	%
+	Therefore $a(X, G, W) + a(X, G, W^{\vee})$ equals:
+	%
+	\begin{align*}
+		2 (g_Y - 1) \cdot \dim_k W 
+		&+ \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} 
+		\frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} 
+		\cdot \big(N_{Q, i}(W) + N_{Q, i}(W^{\vee})\big) \\
+		&\quad + 2 \llbracket W \cong k \rrbracket \\
+		&= 2 (g_Y - 1) \cdot \dim_k W 
+		+ \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} 
+		\left(\frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} 
+		+ \frac{i}{e_{X/Y, Q}}\right) \cdot N_{Q, i}(W) \\
+		&\quad + 2 \llbracket W \cong k \rrbracket \\
+		&= 2 (g_Y - 1) \cdot \dim_k W 
+		+ \sum_{Q \in Y(k)} \big(\dim_k W - \dim_k W^{G_Q}\big) \\
+		&\quad + 2 \llbracket W \cong k \rrbracket.
+	\end{align*}
+	%
+	This ends the proof.
+	%
+\end{proof}
 }
 
 \section{Cyclic covers}
@@ -287,8 +326,7 @@ For any $k[H]$-module $M$ denote:
 	T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n.
 \end{align*}
 %
-Recall that $\dim_k T^i M$, 
-{\color{red} for $i=1, \ldots,p^n$}
+Recall that $\dim_k T^i M$ for $i=1, \ldots, p^n$
 determines the structure of $M$ completely (see \cite[p. 108]{Valentini_Madan_Automorphisms} -- they give the argument for $M := H^0(X, \Omega_X)$,
 but it works for an arbitrary module).
 Moreover, for $i > 0$:
@@ -339,12 +377,9 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.
 		= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k).
 	\end{align*}
 	%
-	Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by 
-	{\color{red}
-	\cite[th. 6.2.2]{Weibel}.
-	}
+	Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by \cite[th. 6.2.2]{Weibel}.
 \end{proof}
-{\color{red}
+%
 \begin{Remark}
 The equality $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ does not hold for non-cyclic groups. For example it is known \cite[cor. II.4.3,th. II.4.4]{MR2035696} that the cohomological ring for the elementary abelian group $\mathbb{F}_p^s$ is given by 
 \[
@@ -356,7 +391,6 @@ The equality $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ does not hold for non-cyclic
 \]
 Therefore, for $s>1$ the degree one and two parts of the cohomological ring, which correspond to the first and second cohomology groups, have different dimensions. 
 \end{Remark}
-}
 %
 \begin{Lemma} \label{lem:trace_surjective}
 	Suppose that $G$ is a $p$-group.
@@ -444,7 +478,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
 %
 \begin{Lemma} \label{lem:u_equals_ul}
-	Assume that $ {\color{red} \phi:} Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
+	Assume that $\phi: Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
 	Then:
 	%
 	\[
@@ -463,11 +497,8 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	\end{equation}
 	%
 	Assume now that $Q \in B_{Y'/Y}$. Then there exists a unique point $Q' \in Y'(k)$
-	in the preimage of $Q$ through ${\color{red} \phi:}Y' \to Y$. Moreover, $m_{X/Y, Q} = n$, $m_{X/Y', Q'} = n-1$. 
-	Recall also that by 
-	{\color{red}
-	\cite[Example p.76]{Serre1979}
-	 }
+	in the preimage of $Q$ through $\phi: Y' \to Y$. Moreover, $m_{X/Y, Q} = n$, $m_{X/Y', Q'} = n-1$. 
+	Recall also that by \cite[Example p.76]{Serre1979}
 	there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that for every $t \ge 0$:
 	%
 	\begin{align*}
@@ -810,7 +841,7 @@ where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref
 $a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and
 %
 \begin{align*}
-	b_W &:= \frac 1p \left( (p-1) \cdot a_W - \sum_{i = 1}^{p-2} a_{W \otimes \chi^i} \right) - a'_{W \otimes \chi}.
+	b_W &:= \frac 1p \left( p \cdot a^{dR}_{X, C}(W) - \sum_{i = 0}^{p-2} a^{dR}_{X, G}(W \otimes \chi^i) \right) - a^{dR}_{Y, G}(W \otimes \chi).
 \end{align*}
 %
 \end{Proposition}