From 262535f385b05cbd388279c79bfdfd1a61848db0 Mon Sep 17 00:00:00 2001
From: jgarnek <jgarnek@amu.edu.pl>
Date: Mon, 9 Dec 2024 15:38:21 +0100
Subject: [PATCH] monday 9.12. pt 1

---
 article_de_rham_cyclic.tex | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex
index 6516376..72ae431 100644
--- a/article_de_rham_cyclic.tex
+++ b/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
 %
 \section{Examples}
 %
+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
 %
 \[