From c44ac3b62056b7d94e5f18344286ba1d056b267d Mon Sep 17 00:00:00 2001 From: jgarnek Date: Thu, 17 Oct 2024 20:22:54 +0200 Subject: [PATCH] pf for hypoelementary --- article_de_rham_cyclic.out | 3 +- article_de_rham_cyclic.synctex.gz | Bin 25714 -> 42884 bytes article_de_rham_cyclic.tex | 175 ++++++++++++++++++++++++++---- 3 files changed, 154 insertions(+), 24 deletions(-) diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out index b97ff67..f94b2b7 100644 --- a/article_de_rham_cyclic.out +++ b/article_de_rham_cyclic.out @@ -1,4 +1,5 @@ \BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 2 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000H\000y\000p\000o\000e\000l\000e\000m\000e\000n\000t\000a\000r\000y\000\040\000c\000o\000v\000e\000r\000s}{}% 3 -\BOOKMARK 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M \cong H^1_{dR}(X'')$ + is also determined by ramification data (???). + + Assume now that $X \to Y$ is not etale. 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: + % + \begin{equation} \label{eqn:TiM=T1M_chi} + T^{i+1} \mc M \cong (T^1 \mc M)^{\chi^{-i}} + \end{equation} + % + for $i \le p^n - p^{n-1}$. Observe that $\mc T^i M$ has the filtration $\mc M^{(pi)} \supset \mc M^{(pi - 1)} \supset \ldots \supset \mc M^{(pi - p)}$ with subquotients $T^{pi} \mc M, \ldots, T^{pi - p + 1} \mc M$. + Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$: + % + \begin{align*} + \mc T^i \mc M &\cong T^{pi - p + 1} \mc M \oplus \ldots \oplus T^{pi} \mc M\\ + &\cong T^1 \mc M \oplus (T^1 \mc M)^{\chi^{-1}} \oplus \ldots \oplus + (T^1 \mc M)^{\chi^{-p}}. + \end{align*} + % + By induction assumption, the $k[C]$-module structure of $\mc T^i \mc M$ is uniquely determined by the ramification data. Thus, by Lemma~\ref{lem:N+Nchi+...} for $N := T^1 \mc M$ and by~\eqref{eqn:TiM=T1M_chi} the $k[C]$-structure of the modules $T^i \mc M$ is uniquely determined by the ramification data for $i \le p^n - p^{n-1}$. + By similar reasoning, $\tr_{X/X'}$ yields an isomorphism: + % + \[ + T^{i + p^n - p^{n-1}} \mc M \cong (\mc T^i \mc M'')^{\chi^{-1??}}. + \] + % + Thus, by induction hypothesis for $\mc M''$, the $k[C]$-structure of $T^{i + p^n - p^{n-1}} \mc M$ + is determined by ramification data as well. +\end{proof} + +\section{Proof of Main Theorem} +% +(Conlon induction ???) (algebraic closure ???) \bibliography{bibliografia} \end{document} \ No newline at end of file