references for Chevalley Weil; H1 and H2 of G for elementary

article_de_rham_cyclic.bbl

@@ -1,125 +1,133 @@
-\begin{thebibliography}{10}
-
-\bibitem{Alperin_local_rep}
-J.~L. Alperin.
-\newblock {\em Local representation theory}, volume~11 of {\em Cambridge
-  Studies in Advanced Mathematics}.
-\newblock Cambridge University Press, Cambridge, 1986.
-\newblock Modular representations as an introduction to the local
-  representation theory of finite groups.
-
-\bibitem{Bleher_Camacho_Holomorphic_differentials}
-F.~M. Bleher and N.~Camacho.
-\newblock Holomorphic differentials of {K}lein four covers.
-\newblock {\em J. Pure Appl. Algebra}, 227(10):Paper No. 107384, 27, 2023.
-
-\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{Borevic_Faddeev}
-Z.~I. {Borevi\v{c}} and D.~K. Faddeev.
-\newblock Theory of homology in groups. {II}. {P}rojective resolutions of
-  finite groups.
-\newblock {\em Vestnik Leningrad. Univ.}, 14(7):72--87, 1959.
-
-\bibitem{Chevalley_Weil_Uber_verhalten}
-C.~Chevalley, A.~Weil, and E.~Hecke.
-\newblock \"{U}ber das verhalten der integrale 1. gattung bei automorphismen
-  des funktionenk\"{o}rpers.
-\newblock {\em Abh. Math. Sem. Univ. Hamburg}, 10(1):358--361, 1934.
-
-\bibitem{Curtis_Reiner_Methods_II}
-C.~W. Curtis and I.~Reiner.
-\newblock {\em Methods of representation theory. {V}ol. {II}}.
-\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc.,
-  New York, 1987.
-\newblock With applications to finite groups and orders, A Wiley-Interscience
-  Publication.
-
-\bibitem{Dummigan_99}
-N.~Dummigan.
-\newblock Complete {$p$}-descent for {J}acobians of {H}ermitian curves.
-\newblock {\em Compositio Math.}, 119(2):111--132, 1999.
-
-\bibitem{Garnek_equivariant}
-J.~Garnek.
-\newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence.
-\newblock {\em Math. Z.}, 300(2):1917--1938, 2022.
-
-\bibitem{Garnek_p_gp_covers}
-J.~Garnek.
-\newblock {$p$}-group {G}alois covers of curves in characteristic {$p$}.
-\newblock {\em Trans. Amer. Math. Soc.}, 376(8):5857--5897, 2023.
-
-\bibitem{Garnek_p_gp_covers_ii}
-J.~Garnek.
-\newblock $p$-group {G}alois covers of curves in characteristic $p$ {II}, 2023.
-
-\bibitem{garnek_indecomposables}
-J.~Garnek.
-\newblock Indecomposable direct summands of cohomologies of curves, 2024.
-\newblock arXiv 2410.03319.
-
-\bibitem{Gross_Rigid_local_systems_Gm}
-B.~H. Gross.
-\newblock Rigid local systems on {$\Bbb G_m$} with finite monodromy.
-\newblock {\em Adv. Math.}, 224(6):2531--2543, 2010.
-
-\bibitem{Hartshorne1977}
-R.~Hartshorne.
-\newblock {\em {Algebraic geometry}}.
-\newblock Springer-Verlag, New York-Heidelberg, 1977.
-\newblock Graduate Texts in Mathematics, No. 52.
-
-\bibitem{Heller_Reiner_Reps_in_integers_I}
-A.~Heller and I.~Reiner.
-\newblock Representations of cyclic groups in rings of integers. {I}.
-\newblock {\em Ann. of Math. (2)}, 76:73--92, 1962.
-
-\bibitem{Higman}
-D.~G. Higman.
-\newblock Indecomposable representations at characteristic {$p$}.
-\newblock {\em Duke Math. J.}, 21:377--381, 1954.
-
-\bibitem{laurent_kock_drinfeld}
-L.~Laurent and B.~K{\"{o}}ck.
-\newblock The canonical representation of the drinfeld curve.
-\newblock {\em Mathematische Nachrichten}, online first, 2024.
-
-\bibitem{Lusztig_Coxeter_orbits}
-G.~Lusztig.
-\newblock Coxeter orbits and eigenspaces of {F}robenius.
-\newblock {\em Invent. Math.}, 38(2):101--159, 1976/77.
-
-\bibitem{WardMarques_HoloDiffs}
-S.~Marques and K.~Ward.
-\newblock Holomorphic differentials of certain solvable covers of the
-  projective line over a perfect field.
-\newblock {\em Math. Nachr.}, 291(13):2057--2083, 2018.
-
-\bibitem{Prest}
-M.~Prest.
-\newblock Wild representation type and undecidability.
-\newblock {\em Comm. Algebra}, 19(3):919--929, 1991.
-
-\bibitem{Serre1979}
-J.-P. Serre.
-\newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in
-  Mathematics}}.
-\newblock Springer-Verlag, New York-Berlin, 1979.
-\newblock Translated from the French by Marvin Jay Greenberg.
-
-\bibitem{Steinberg_Representation_book}
-B.~Steinberg.
-\newblock {\em Representation theory of finite groups}.
-\newblock Universitext. Springer, New York, 2012.
-\newblock An introductory approach.
-
-\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}
+ \def\cprime{$'$}
+\begin{thebibliography}{10}
+
+\bibitem{MR2035696}
+Alejandro Adem and R.~James Milgram.
+\newblock {\em Cohomology of finite groups}, volume 309 of {\em Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}.
+\newblock Springer-Verlag, Berlin, second edition, 2004.
+
+\bibitem{Alperin_local_rep}
+J.~L. Alperin.
+\newblock {\em Local representation theory}, volume~11 of {\em Cambridge Studies in Advanced Mathematics}.
+\newblock Cambridge University Press, Cambridge, 1986.
+\newblock Modular representations as an introduction to the local representation theory of finite groups.
+
+\bibitem{Bleher_Camacho_Holomorphic_differentials}
+F.~M. Bleher and N.~Camacho.
+\newblock Holomorphic differentials of {K}lein four covers.
+\newblock {\em J. Pure Appl. Algebra}, 227(10):Paper No. 107384, 27, 2023.
+
+\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{Borevic_Faddeev}
+Z.~I. {Borevi\v{c}} and D.~K. Faddeev.
+\newblock Theory of homology in groups. {II}. {P}rojective resolutions of finite groups.
+\newblock {\em Vestnik Leningrad. Univ.}, 14(7):72--87, 1959.
+
+\bibitem{Chevalley_Weil_Uber_verhalten}
+C.~Chevalley, A.~Weil, and E.~Hecke.
+\newblock \"{U}ber das verhalten der integrale 1. gattung bei automorphismen des funktionenk\"{o}rpers.
+\newblock {\em Abh. Math. Sem. Univ. Hamburg}, 10(1):358--361, 1934.
+
+\bibitem{Curtis_Reiner_Methods_II}
+C.~W. Curtis and I.~Reiner.
+\newblock {\em Methods of representation theory. {V}ol. {II}}.
+\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc., New York, 1987.
+\newblock With applications to finite groups and orders, A Wiley-Interscience Publication.
+
+\bibitem{Dummigan_99}
+N.~Dummigan.
+\newblock Complete {$p$}-descent for {J}acobians of {H}ermitian curves.
+\newblock {\em Compositio Math.}, 119(2):111--132, 1999.
+
+\bibitem{Ellingsrud_Lonsted_Equivariant_Lefschetz}
+G.~Ellingsrud and K.~L\o~nsted.
+\newblock An equivariant {L}efschetz formula for finite reductive groups.
+\newblock {\em Math. Ann.}, 251(3):253--261, 1980.
+
+\bibitem{Garnek_equivariant}
+J.~Garnek.
+\newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence.
+\newblock {\em Math. Z.}, 300(2):1917--1938, 2022.
+
+\bibitem{Garnek_p_gp_covers}
+J.~Garnek.
+\newblock {$p$}-group {G}alois covers of curves in characteristic {$p$}.
+\newblock {\em Trans. Amer. Math. Soc.}, 376(8):5857--5897, 2023.
+
+\bibitem{Garnek_p_gp_covers_ii}
+J.~Garnek.
+\newblock $p$-group {G}alois covers of curves in characteristic $p$ {II}, 2023.
+
+\bibitem{garnek_indecomposables}
+J.~Garnek.
+\newblock Indecomposable direct summands of cohomologies of curves, 2024.
+\newblock arXiv 2410.03319.
+
+\bibitem{Gross_Rigid_local_systems_Gm}
+B.~H. Gross.
+\newblock Rigid local systems on {$\Bbb G_m$} with finite monodromy.
+\newblock {\em Adv. Math.}, 224(6):2531--2543, 2010.
+
+\bibitem{Hartshorne1977}
+R.~Hartshorne.
+\newblock {\em {Algebraic geometry}}.
+\newblock Springer-Verlag, New York-Heidelberg, 1977.
+\newblock Graduate Texts in Mathematics, No. 52.
+
+\bibitem{Heller_Reiner_Reps_in_integers_I}
+A.~Heller and I.~Reiner.
+\newblock Representations of cyclic groups in rings of integers. {I}.
+\newblock {\em Ann. of Math. (2)}, 76:73--92, 1962.
+
+\bibitem{Higman}
+D.~G. Higman.
+\newblock Indecomposable representations at characteristic {$p$}.
+\newblock {\em Duke Math. J.}, 21:377--381, 1954.
+
+\bibitem{laurent_kock_drinfeld}
+L.~Laurent and B.~K{\"{o}}ck.
+\newblock The canonical representation of the drinfeld curve.
+\newblock {\em Mathematische Nachrichten}, online first, 2024.
+
+\bibitem{Lusztig_Coxeter_orbits}
+G.~Lusztig.
+\newblock Coxeter orbits and eigenspaces of {F}robenius.
+\newblock {\em Invent. Math.}, 38(2):101--159, 1976/77.
+
+\bibitem{WardMarques_HoloDiffs}
+S.~Marques and K.~Ward.
+\newblock Holomorphic differentials of certain solvable covers of the projective line over a perfect field.
+\newblock {\em Math. Nachr.}, 291(13):2057--2083, 2018.
+
+\bibitem{Prest}
+M.~Prest.
+\newblock Wild representation type and undecidability.
+\newblock {\em Comm. Algebra}, 19(3):919--929, 1991.
+
+\bibitem{Serre1979}
+J.-P. Serre.
+\newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in Mathematics}}.
+\newblock Springer-Verlag, New York-Berlin, 1979.
+\newblock Translated from the French by Marvin Jay Greenberg.
+
+\bibitem{Steinberg_Representation_book}
+B.~Steinberg.
+\newblock {\em Representation theory of finite groups}.
+\newblock Universitext. Springer, New York, 2012.
+\newblock An introductory approach.
+
+\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.
+
+\bibitem{Weibel}
+Ch.~A. Weibel.
+\newblock {\em An introduction to homological algebra}, volume~38 of {\em Cambridge Studies in Advanced Mathematics}.
+\newblock Cambridge University Press, Cambridge, 1994.
+
+\end{thebibliography}

article_de_rham_cyclic.tex

@@ -132,14 +132,21 @@ hyperref, bbm, mathtools, mathrsfs}
 %
 \section{Introduction}
 %
-The classical Chevalley--Weil formula (cf. \cite{Chevalley_Weil_Uber_verhalten}) gives an explicit description
+The classical Chevalley--Weil formula 
+(cf. \cite{Chevalley_Weil_Uber_verhalten}, 
+{\color{red} 
+\cite{Ellingsrud_Lonsted_Equivariant_Lefschetz})
+}
+ gives an explicit description
 of the equivariant structure of the cohomology of a curve $X$ with a group action over a field of characteristic~$0$. Their formula depends on the so-called \emph{fundamental characters} of points $x \in X$ that are ramified in the cover $X \to X/G$. ????
 
 It is hard to expect such a formula over fields of characteristic~$p$.
 Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. \cite{Prest}). There are many results concerning equivariant structure of cohomologies  for particular groups
 (see e.g.~\cite{Valentini_Madan_Automorphisms} for the case of cyclic groups, \cite{WardMarques_HoloDiffs} for abelian groups, \cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} for groups with a cyclic Sylow subgroup, or \cite{Bleher_Camacho_Holomorphic_differentials} for the Klein group) or curves (cf. \cite{Lusztig_Coxeter_orbits}, \cite{Dummigan_99}, \cite{Gross_Rigid_local_systems_Gm}, \cite{laurent_kock_drinfeld}). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. \cite{Garnek_p_gp_covers}, \cite{Garnek_p_gp_covers_ii}). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\
 
-This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of indecomposable modules is finite (cf. \cite{Higman}, \cite{Borevic_Faddeev}, \cite{Heller_Reiner_Reps_in_integers_I}). While their representation theory still
+This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of 
+{\color{red} equivalence classes of }
+indecomposable modules is finite (cf. \cite{Higman}, \cite{Borevic_Faddeev}, \cite{Heller_Reiner_Reps_in_integers_I}). While their representation theory still
 seems a bit too complicated to derive a general formula for the cohomologies, 
 the article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} proved that 
 the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data (i.e. higher ramification groups and the fundamental characters of the ramification locus). The main result of this article is a similar statement for the de Rham cohomology.
@@ -148,7 +155,7 @@ the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ra
 	Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
 	Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
 	The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the
-	higher ramification data of the cover $X \to X/G$ and the genus of $X/G$. ??genus of X/G??
+	higher ramification data of the cover $X \to X/G$ and the genus of $X$.
 \end{mainthm}
 %
 Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
@@ -223,7 +230,9 @@ For any $k[H]$-module $M$ denote:
 	T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n.
 \end{align*}
 %
-Recall that $\dim_k T^i M$ determines the structure of $M$ completely (see \cite[p. 108]{Valentini_Madan_Automorphisms} -- they give the argument for $M := H^0(X, \Omega_X)$,
+Recall that $\dim_k T^i M$, 
+{\color{red} for $i=1, \ldots,p^n$}
+determines the structure of $M$ completely (see \cite[p. 108]{Valentini_Madan_Automorphisms} -- they give the argument for $M := H^0(X, \Omega_X)$,
 but it works for an arbitrary module).
 Moreover, for $i > 0$:
 %
@@ -284,8 +293,23 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.
 		= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k).
 	\end{align*}
 	%
-	Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by ????.
+	Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by 
+	{\color{red}
+	\cite[th. 6.2.2]{Weibel}.
+	}
 \end{proof}
+{\color{red}
+\begin{Remark}
+The equality $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ does not hold for non-cyclic groups. For example it is known \cite[cor. II.4.3,th. II.4.4]{MR2035696} that the cohomological ring for the elementary abelian group $\mathbb{F}_p^s$ is given by 
+\[
+	H^* (G, \mathbb{F}_p)=
+	\begin{cases}
+		\mathbb{F}_2[x_1, \ldots,x_s] & \text{ if } p=2 \\
+		\wedge(x_{1}, \ldots, x_s) \otimes \mathbb{F}_p[x_1, \ldots,x_s] & \text{ if } p>2
+	\end{cases}
+\]
+\end{Remark}
+}
 %
 \begin{Lemma} \label{lem:trace_surjective}
 	Suppose that $G$ is a $p$-group.
@@ -799,5 +823,7 @@ Basis of holomorphic differentials:
 }
 
 
-\bibliography{bibliografia}
+\bibliography{bibliografia,AKGeneral}
+% 
+% \bibliography{AKGeneral}
 \end{document}