of the equivariant structure of the cohomology of a curve $X$ with a group action over a field of characteristic~$0$. Their formula depends on the so-called \emph{fundamental characters} of points $x \in X$ that are ramified in the cover $X \to X/G$. ????
It is hard to expect such a formula over fields of characteristic~$p$.
Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. \cite{Prest}). There are many results concerning equivariant structure of cohomologies for particular groups
(see e.g.~\cite{Valentini_Madan_Automorphisms} for the case of cyclic groups, \cite{WardMarques_HoloDiffs} for abelian groups, \cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} for groups with a cyclic Sylow subgroup, or \cite{Bleher_Camacho_Holomorphic_differentials} for the Klein group) or curves (cf. \cite{Lusztig_Coxeter_orbits}, \cite{Dummigan_99}, \cite{Gross_Rigid_local_systems_Gm}, \cite{laurent_kock_drinfeld}). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. \cite{Garnek_p_gp_covers}, \cite{Garnek_p_gp_covers_ii}). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\
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 still
seems a bit too complicated to derive a general formula for the cohomologies,
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.
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 varying $k[G]$-module structure of $H^0(X, \Omega_X)$ and~$H^1_{dR}(X)$.\\
We elaborate now on the proof of Main Theorem. The first step is to prove Main Theorem for $G =\ZZ/p^n$. In this case we give an explicit formula for the structure of $H^1_{dR}(X)$, depending only on the ramification indices, ramification jumps and genus of the quotient
curve (see Theorem~\ref{thm:cyclic_de_rham}). This formula is proven inductively, by applying induction twice: once for the curve $X$ with an action of $\ZZ/p^{n-1}$ and once for the curve $X'' := X/(\ZZ/p)$.
In the second step, we use similar methods to show the result for a group of the form
$\ZZ/p^n \rtimes_{\chi}\ZZ/c$. Finally, we use Conlon induction theorem to deduce Main Theorem
for an arbitrary group with a cyclic $p$-Sylow subgroup.
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\section{Notation}
%
Assume that $\pi : X \to Y$ is a $G$-cover of smooth projective curves over an field $k$
of characteristic $p$.
Throughout the paper we will use the following notation for any $P \in X(\ol k)$:
\begin{itemize}
\item$e_{X/Y, P}$ is the ramification index at $P$,
Note that for $g_Y =0$ the first exponent is negative. However, since $m_{X/Y}= n$ (as $\PP^1$ doesn't have any \'{e}tale covers), the first two summands in~\eqref{eqn:HdR_formula} cancel out. Thus also in this case the module~\eqref{eqn:HdR_formula} is well-defined.
Recall that $\dim_k T^i M$ determines the structure of $M$ completely (see \cite[p. 108]{Valentini_Madan_Automorphisms} -- they give the argument for $M := H^0(X, \Omega_X)$,
Let $\HH^i(Y, \mc F^{\bullet})$ be the $i$th hypercohomology of a complex $\mc F^{\bullet}$.
Write also $\mc H^i(G, -)$ for the $i$th derived functor of the functor
%
\[
\mc F \mapsto\mc F^G.
\]
%
Since $X \to Y$ is \'{e}tale, $\mc H^i(G, \pi_*\mc F)=0$ for any $i > 0$ and any coherent sheaf $\mc F$ on $X$ by \cite[Proposition~2.1]{Garnek_equivariant}.
Therefore the spectral sequence~\cite[(3.4)]{Garnek_equivariant} applied for the complex $\mc F^{\bullet} :=\pi_*\Omega_{X/k}^{\bullet}$ yields $\RR^i \Gamma^G(\pi_*\Omega_{X/k}^{\bullet})=\HH^1(Y, \pi_*^G \Omega_{X/k}^{\bullet})= H^1_{dR}(Y)$, since $\pi_*^G \Omega_X^{\bullet}\cong\Omega_Y$ (cf. ???).
On the other hand, the seven-term exact sequence applied for the spectral sequence~\cite[(3.5)]{Garnek_equivariant} yields:
where the rows are Hodge--de Rham exact sequences. Recall that by~\cite[Theorem~1]{Valentini_Madan_Automorphisms}, in this case $H^0(X, \Omega_X)$ contains
a copy of $k[G]^{\oplus g_Y}$ as a direct summand. Thus, since trace is injective on $k[G]^{\oplus g_Y}$, the dimension
of the image of
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\begin{equation}\label{eqn:trace_H0_Omega}
\tr_{X/Y} : H^0(X, \Omega_X) \to H^0(Y, \Omega_Y)
\end{equation}
%
is $g_Y$. Therefore the map~\eqref{eqn:trace_H0_Omega} is surjective.
Similarly, by Serre's duality, also $H^1(X, \mc O_X)$ contains $k[G]^{\oplus g_Y}$ as a direct summand
where $e'_Q := e_{X/Y', Q}$ and $Q_1\in\pi^{-1}(Q_0)$. Therefore, for $i \le p^{n-1}- p^{n-2}$, using the Riemann--Hurwitz formula (cf. \cite[Corollary~IV.2.4]{Hartshorne1977}) and Lemma~\ref{lem:u_equals_ul}:
By Lemma~\ref{lem:trace_surjective} since $X \to X''$ is not \'{e}tale, the map $\tr_{X/X''} : H^1_{dR}(X)\to H^1_{dR}(X'')$ is surjective. Recall that
and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{j + p^n - p^{n-1}}\mc M \to\mc T^j \mc M''$ for any $j \ge1$. ?? Therefore, if $i \in(p^n - p^{N+1}, p^n - p^N]$:
Suppose $M$ is a finitely generated $k[G]$-module.
\begin{enumerate}[leftmargin=*]
\item The $k[G]$-module structure of $M$
is uniquely determined by the restrictions $M|_H$ as $H$ ranges over all $p$-hypo-elementary subgroups of $G$.
\item The $k[G]$-module structure of $M$ is uniquely determined by the $\ol k[G]$-module structure of $M \otimes_k \ol k$.
\end{enumerate}
\end{Lemma}
\begin{proof}
\begin{enumerate}[leftmargin=*]
\item This follows easily from Conlon induction theorem (cf. \cite[Theorem~(80.51)]{Curtis_Reiner_Methods_II}), see e.g. \cite[Lemma~3.2]{Bleher_Chinburg_Kontogeorgis_Galois_structure}.
\item This is \cite[Proposition~3.5. (iii)]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
By Lemma~\ref{lem:reductions} we may assume that $G = H \rtimes_{\chi} C =\langle\sigma\rangle\rtimes_{\chi}\langle\rho\rangle\cong\ZZ/p^n \rtimes_{\chi}\ZZ/c$ and that $k$ is algebraically closed.
Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-module $M$ and any character $\psi$ of $H$ we write $M^{\psi} := M \otimes_{k[C]}\psi$.
This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Recall that if $U$ is an indecomposable $k[G]$-module
then $U^{\sigma} :=\ker(\sigma-1)$ (the socle of $U$) is a one-dimensional
$k[C]$-module. Thus it comes from a character $\chi_U \in\wh{C} :=\Hom(C, \CC)$.
It turns out that the map
%
\[
U \mapsto (\dim_k U, \chi_U)
\]
%
is a bijection between the set of indecomposable $k[G]$-modules and the set $\{1, \ldots, p^n -1\}\times\wh{C}$. Fix a character $\chi$ that generates $\wh{C}$.
Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$
and dimension $b$. Write
%
\[
M \cong\bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}.
As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C =\langle\sigma\rangle\rtimes_{\chi}\langle\rho\rangle\cong\ZZ/p^n \rtimes_{\chi}\ZZ/c$ and $k \ol k$ by Lemma~\ref{lem:reductions}.
We prove this by induction on $n$. If $n =0$, then it follows by Chevalley--Weil theorem.
Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale.
Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma-1)$
is an isomorphism of $k$-vector spaces between $T^{i+1}\mc M$ and $T^i \mc M$ for
$i =2, \ldots, p^n$. This yields an isomorphism of $k[C]$-modules for $i \ge2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}:
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\begin{equation}\label{eqn:TiM=T1M_chi_\'{e}tale}
T^i \mc M \cong (T^2 \mc M)^{\chi^{-i+2}}
\end{equation}
%
Observe that $\mc T^i \mc M$ has the filtration $\mc M^{(pi)}\supset\mc M^{(pi -1)}\supset\ldots\supset\mc M^{(pi - p)}$ with subquotients $T^{pi}\mc M, \ldots, T^{pi - p}\mc M$.
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:
%
\begin{equation}\label{eqn:TiM=T1M_chi}
T^{i+1}\mc M \cong (T^1 \mc M)^{\chi^{-i}}
\end{equation}
%
for $i \le p^n - p^{n-1}$. Observe that $\mc T^i M$ has the filtration $\mc M^{(pi)}\supset\mc M^{(pi -1)}\supset\ldots\supset\mc M^{(pi - p)}$ with subquotients $T^{pi}\mc M, \ldots, T^{pi - p +1}\mc M$.
Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$:
%
\begin{align*}
\mc T^i \mc M &\cong T^{pi - p + 1}\mc M \oplus\ldots\oplus T^{pi}\mc M\\
&\cong T^1 \mc M \oplus (T^1 \mc M)^{\chi^{-1}}\oplus\ldots\oplus
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 $i \le p^n - p^{n-1}$.
