Ti M determines structure of M

article_de_rham_cyclic.tex

@@ -231,7 +231,7 @@ $\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k
 Finally, we recall the classical Chevalley-Weil formula. Keep the above notation and assume that $p \nmid \# G$. For any $Q \in Y(k)$ let $\chi_Q : G_Q \to k^{\times}$ be the fundamental character of $G_Q$ acting on the tangent space of $Q$. Then:
 %
 \begin{equation}
-	H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} M^{\oplus a_M},
+	H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M},
 \end{equation}
 %
 where:
@@ -630,8 +630,33 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Let $\psi : C \to k^{\times}$ be a primitive character. Write
 	%
 	\[
-		M \cong \bigoplus_{a, b} \mc V(\psi^a, b)^{\oplus n(a, b)}.
+		M \cong \bigoplus_{i = 1}^{p^n} \bigoplus_{W \in \Indec(C)} \mc V(W, i)^{\oplus n(W, i)}.
 	\]
+	%
+	Note that as $k[C]$-modules:
+	%
+	\[
+		T^j \mc V(W, i) \cong
+		\begin{cases}
+			W^{\chi^{-j + 1}}, & \textrm{ if } j \le i,\\
+			0, & \textrm{ if } j > i.
+		\end{cases}
+	\]
+	%
+	Hence:
+	%
+	\[
+	T^j M \cong \bigoplus_{i = j}^{p^n} \bigoplus_{W \in \Indec(C)} (W^{\chi^{-j + 1}})^{\oplus n(W, i)}
+	\]
+	%
+	and the $k[C]$-module structure of $T^j M$ determines uniquely 
+	the numbers:
+	%
+	\[
+		\sum_{i = j}^{p^n} n(W, i)
+	\]
+	%
+	for every $W \in \Indec(k[C])$. This easily implies that the numbers $n(W, 1)$, $\ldots$, $n(W, p^n)$ are uniquely determined by the $k[C]$-structure of $T^1 M$, $\ldots$, $T^{p^n} M$. The proof follows.
 \end{proof}
 %
 \begin{Lemma} \label{lem:N+Nchi+...}