monday 9.12. pt 1

article_de_rham_cyclic.tex

@@ -753,6 +753,22 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
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 \section{Examples}
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+Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order $p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$
+for a homomorphism $\chi : C \to \FF_p^{\times}$. 
+%
+\begin{Proposition}
+Keep the above notation. If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then:
+%
+\[
+	H^1_{dR}(X) \cong \bigoplus_{M \in \Indec(C)} \mc V(M, p-1)^{\oplus b_M} \oplus \mc V(M, p)^{\oplus c_M},
+\]	
+%
+where $b_M := \ldots$, $c_M := \ldots$.
+\end{Proposition}
+\begin{proof}
+	???
+\end{proof}
+
 \noindent Let $p > 2$. Consider the Mumford curve
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