diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl
index 334e3eb..f4518a9 100644
--- a/article_de_rham_cyclic.bbl
+++ b/article_de_rham_cyclic.bbl
@@ -1,5 +1,13 @@
 \begin{thebibliography}{10}
 
+\bibitem{Alperin_local_rep}
+J.~L. Alperin.
+\newblock {\em Local representation theory}, volume~11 of {\em Cambridge
+  Studies in Advanced Mathematics}.
+\newblock Cambridge University Press, Cambridge, 1986.
+\newblock Modular representations as an introduction to the local
+  representation theory of finite groups.
+
 \bibitem{Bleher_Camacho_Holomorphic_differentials}
 F.~M. Bleher and N.~Camacho.
 \newblock Holomorphic differentials of {K}lein four covers.
diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz
index e06555e..47a6ef8 100644
Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ
diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex
index d17e436..c979f5d 100644
--- a/article_de_rham_cyclic.tex
+++ b/article_de_rham_cyclic.tex
@@ -560,9 +560,20 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$.
 \end{Lemma}
 \begin{proof}
-	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 $\chi_U \in \wh{C} := \Hom(C, \CC)$.
+	This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. 
+	Assume that $G$ is a finite group with a normal cyclic $p$-Sylow subgroup $H = \langle \sigma \rangle \cong \ZZ/p^n$. Let $C := G/H$.
+	Recall that if $U$ is an indecomposable $k[G]$-module
+	then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
+	$k[C]$-module. It turns out that the map
+	%
+	\begin{align*}
+		\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
+		U \mapsto (U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}})
+	\end{align*}
+	%
+	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 $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
+	Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$.
 	It turns out that the map
 	%
 	\[
@@ -672,7 +683,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	%
 	\begin{align}
 		\mc T^1 \mc M &\cong T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}} \label{eqn:decomposition_of_mc_T1}\\
-		\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^n - p^{n-1}. \label{eqn:decomposition_of_mc_Ti}
+		\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^{n-1}. ???? \label{eqn:decomposition_of_mc_Ti}
 	\end{align}
 	%
 	Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by higher ramification data,