lematy

article_de_rham_cyclic.bbl

@@ -1,3 +1,13 @@
-\begin{thebibliography}{}
+\begin{thebibliography}{1}
+
+\bibitem{Bleher_Chinburg_Kontogeorgis_Galois_structure}
+F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis.
+\newblock Galois structure of the holomorphic differentials of curves.
+\newblock {\em J. Number Theory}, 216:1--68, 2020.
+
+\bibitem{Valentini_Madan_Automorphisms}
+R.~C. Valentini and M.~L. Madan.
+\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}.
+\newblock {\em J. Number Theory}, 13(1):106--115, 1981.
 
 \end{thebibliography}

article_de_rham_cyclic.tex

@@ -202,6 +202,7 @@ Note also that for $j \ge 1$:
 \end{proof}
 %
 \begin{Lemma} \label{lem:trace_surjective}
+	Suppose that $G$ is a $p$-group.
 	If the $G$-cover $X \to Y$ is totally ramified, then the map
 	%
 	\[
@@ -211,18 +212,58 @@ Note also that for $j \ge 1$:
 	is an epimorphism.
 \end{Lemma}
 \begin{proof}
-	????
+%
+By induction, it suffices to prove this in the case when $G = \ZZ/p$.
+Consider the following commutative diagram:
+%
+\begin{center}
+	% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKMgEYqtRizYduvbHgJFh5MfWatEIABIA9YgAoAGqQAEAHVMB5ALYwA5nQD6BgJRceIDHIGLSo6qskNHX0ATRNzaztHENcZDz55QWQAJmV-CXUtbWEHYCgAJU5DWPdPfgUUVL9xNTYdHLzCvRi3WXKkgGY0msCs4UNw0ysAY2MLJxK2xKIu6oDM+ubBkbGHFriy6ZQAFm75qVb4rwrkXbmMg84xGChbeCJQADMAJwgrJDIQHAgkZLiXt6-ajfJDbf6vd6IXZfH6IABs4MB8OBsIAHIjIQB2FFIACcGKQAFYcYhMQTEF0YUTyUoqRTyak6ZT9hpzDhnrkDAB6EKcQ4AyHQkGIYk9TJsjnAbm8-kQpBwknYsVsCWcnl8q6cIA
+	\begin{tikzcd}
+		0 \arrow[r] & {H^0(X, \Omega_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & H^1_{dR}(X) \arrow[r] \arrow[d, "\tr_{X/Y}"] & {H^1(X, \mc O_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & 0 \\
+		0 \arrow[r] & {H^0(Y, \Omega_Y)} \arrow[r]                        & H^1_{dR}(Y) \arrow[r]                        & {H^1(Y, \mc O_Y)} \arrow[r]                        & 0
+	\end{tikzcd}
+\end{center}
+%
+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
+%
+\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
+and one shows similarly that the trace map
+%
+\begin{equation*} %\label{eqn:trace_H0_Omega}
+	\tr_{X/Y} : H^1(X, \mc O_X) \to H^1(Y, \mc O_Y)
+\end{equation*}
+%
+is surjective. Therefore, since the outer trace maps in the diagram are surjective,
+the trace map on the de Rham cohomology must be surjective as well.
+%
 \end{proof}
 %
 \begin{Lemma} \label{lem:TiM_isomorphism}
 	For any $i \le p^n - 1$ we have the following $k$-linear monomorphism:
 	%
 	\[
-		(\sigma - 1) : T^{i+1} M \hookrightarrow T^i M.
+		m_{\sigma - 1} : T^{i+1} M \hookrightarrow T^i M.
 	\]
 \end{Lemma}
 \begin{proof}
-	
+%
+We define $m_{\sigma - 1}$ as follows:
+%
+\[
+	m_{\sigma - 1}(\ol x) := (\sigma - 1) \cdot x,
+\]
+%
+where for $\ol x \in T^i M$ we picked any representative $x \in M^{(i)}$.
+If $x \in M^{(i+1)} := \ker((\sigma - 1)^{i+1})$ then clearly $(\sigma - 1) x \in M^{(i)}$.
+Moreover $(\sigma - 1) \cdot x \in M^{(i-1)}$ holds if and only if $x \in M^{(i)}$. This
+shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
 %
 \begin{Lemma} \label{lem:lemma_mcT_and_T}
@@ -346,7 +387,7 @@ Note also that for $j \ge 1$:
 \section{Hypoelementary covers}
 %
 Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$.
-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 $M^{\psi} := M \otimes_{k[G]} \psi$.
+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 $M^{\psi} := M \otimes_{k[C]} \psi$.
 %
 \begin{Proposition} \label{prop:main_thm_for_hypoelementary}
 	Main Theorem holds for a hypoelementary $G$ as above and $k = \ol k$.
@@ -357,33 +398,84 @@ 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}
-	???
+	See \cite[????]{Bleher_Chinburg_Kontogeorgis_Galois_structure} for a proof.
 \end{proof}
 %
 \begin{Lemma} \label{lem:N+Nchi+...}
-	Let $N_1$, $N_2$ be $k[G]$-modules. Assume that for some $j$
+	Keep the above notation. Let $M$, $N$ be $k[C]$-modules. Assume that
 	%
 	\[
-		N_1 \oplus N_1^{\chi} \oplus \ldots \oplus N_1^{\chi^j} 
+		M \cong N \oplus N^{\chi} \oplus \ldots \oplus N^{\chi^{p-1}}.
-		\cong N_2 \oplus N_2^{\chi} \oplus \ldots \oplus N_2^{\chi^j}.
 	\]
 	%	
-	If $\GCD(j, p-1) = 1$, then $N_1 \cong N_2$.
+	Then $N$ is uniquely determined by $M$.
 	%If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$.
 \end{Lemma}
 \begin{proof}
-	
+	Note that
+	%
+	\[
+		M \cong N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}.
+	\]
+	%
+	By tensoring this isomorphism by $\chi^i$ we obtain:
+	%
+	\begin{align*}
+		M^{\chi^i} \cong (N^{\chi^i})^{\oplus 2} \oplus N^{\chi^{i+1}} \oplus N^{\chi^{i+2}} \oplus \ldots \oplus N^{\chi^{i + p-2}}
+		\cong (N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j}
+	\end{align*}
+	%
+	for $i = 0, \ldots, p-2$. Therefore:
+	%
+	\begin{equation} \label{eqn:N+M=M}
+		N^{\oplus p} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \oplus 
+		\cong M^{\oplus (p-1)}.
+	\end{equation}
+	%
+	Indeed, for the proof of~\eqref{eqn:N+M=M} note that
+	%
+	\begin{align*}
+		N^{\oplus p} &\oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}}
+		\cong N^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((N^{\chi^i})^{\oplus 2}
+		\oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \right)\\
+		&\cong \left( N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} \right)^{\oplus (p-1)}
+		\cong M^{\oplus (p-1)}.
+	\end{align*}
+	%
+	The isomorphism~\eqref{eqn:N+M=M} clearly proves the thesis.
 \end{proof}
 %
 \begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary}
-	For any $i \le p^n - 1$:
+	For any $i \le p^n - 1$ the map~$m_{\sigma - 1}$ from Lemma~\ref{lem:TiM_isomorphism}
+	yields a $k[C]$-equivariant monomorphism:
 	%
 	\[
-	(\sigma - 1) : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}.
+	m_{\sigma - 1} : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}.
 	\]
 \end{Lemma}
 \begin{proof}
-	
+	By Lemma~\ref{lem:TiM_isomorphism} this map is injective. Thus it suffices to check that it is $k[C]$-equivariant.
+	Note that we have the following identity in the ring~$k[C]$:
+	%
+	\[
+		(\sigma - 1) \cdot \rho = \rho \cdot (\sigma^{\chi(\rho)^{-1}} - 1)
+		= \rho \cdot (\sigma - 1) \cdot (1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1})
+	\]
+	%
+	Note that $\sigma$ acts trivially on $T^i M$, so that for any $\ol x \in T^i M$:
+	%
+	\[
+		(1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1}) \cdot \ol x = \chi(\rho)^{-1} \cdot \ol x.
+	\]
+	%
+	This easily shows that
+	%
+	\[
+		m_{\sigma - 1}(\rho \cdot \ol x) = \chi(\rho)^{-1} \cdot \rho \cdot m_{\sigma - 1}(\ol x),
+	\]
+	%
+	which ends the proof.
+	%
 \end{proof}
 
 \begin{proof}[Proof of Proposition~\ref{prop:main_thm_for_hypoelementary}]
@@ -403,15 +495,15 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	\begin{align*}
 		\mc T^i \mc M &\cong
 		\begin{cases}
-			T^1 \mc M \oplus T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-p + 1}}, & i = 1\\
+			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)}}, & i = 1\\
-			T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-p}}, & i > 1.
+			T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}}, & i > 1.
 		\end{cases}
 	\end{align*}
 	%
 	Thus, since by induction hypothesis $\mc T^2 \mc M$ is determined by ramification data,
 	we have by Lemma~\ref{lem:N+Nchi+...} that $T^2 \mc M$ is determined by ramification data.
 	Moreover, by Lemma~\ref{lem:G_invariants_\'{e}tale} and induction hypothesis, $T^1 \mc M \cong H^1_{dR}(X'')$
-	is also determined by ramification data (???).
+	is also determined by ramification data.
 
 	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: