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t?w|ki@Bj99xRl%c-LKq5`fq>x!!LjO>)-$VU*Ep_{{iP+4%$=b2mq`r7~lW^ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 81b6c59..610e717 100644 --- a/article_de_rham_cyclic.tex +++ b/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}