diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index f4518a9..fba4b69 100644 --- a/article_de_rham_cyclic.bbl +++ b/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} diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 47a6ef8..7c54e74 100644 Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 16c188f..e87ea0b 100644 --- a/article_de_rham_cyclic.tex +++ b/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} \ No newline at end of file