Comparisons of Lie algebra cohomologies of $(\varphi ,\Gamma )$-modules
Rustam Steingart1
1 Ruprecht-Karls-Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 665-690

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between $\mathbb{Q}_p$-analytic and $L$-analytic Lie algebra cohomology of analytic $(\varphi _L,\Gamma _L)$-modules. We use the results to conclude that for $L\ne \mathbb{Q}_p,$ there exist examples of étale $(\varphi _L,\Gamma _L)$-modules over Robba rings whose $\mathbb{Q}_p$-analytic cohomology does not arise as a base change of Galois cohomology.

Nous généralisons un résultat de Fourquaux et Xie, en déterminant ainsi complètement la relation entre la cohomologie de l’algèbre de Lie $\mathbb{Q}_p$-analytique et la cohomologie de l’algèbre de Lie $L$-analytique des $(\varphi _L,\Gamma _L)$-modules analytiques. Nous utilisons ces résultats pour conclure que, pour $L\ne \mathbb{Q}_p,$ il existe des exemples de $(\varphi _L,\Gamma _L)$-modules étales sur les anneaux de Robba dont la cohomologie $\mathbb{Q}_p$-analytique ne provient pas d’un changement de base de la cohomologie galoisienne.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1336
Classification : 11F80, 11S25, 22E35
Keywords: $(\varphi ,\Gamma )$-modules, Galois cohomology, analytic cohomology

Rustam Steingart 1

1 Ruprecht-Karls-Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2025__37_2_665_0,
     author = {Rustam Steingart},
     title = {Comparisons of {Lie} algebra cohomologies of $(\varphi ,\Gamma )$-modules},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {665--690},
     year = {2025},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {37},
     number = {2},
     doi = {10.5802/jtnb.1336},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1336/}
}
TY  - JOUR
AU  - Rustam Steingart
TI  - Comparisons of Lie algebra cohomologies of $(\varphi ,\Gamma )$-modules
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2025
SP  - 665
EP  - 690
VL  - 37
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1336/
DO  - 10.5802/jtnb.1336
LA  - en
ID  - JTNB_2025__37_2_665_0
ER  - 
%0 Journal Article
%A Rustam Steingart
%T Comparisons of Lie algebra cohomologies of $(\varphi ,\Gamma )$-modules
%J Journal de théorie des nombres de Bordeaux
%D 2025
%P 665-690
%V 37
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1336/
%R 10.5802/jtnb.1336
%G en
%F JTNB_2025__37_2_665_0
Rustam Steingart. Comparisons of Lie algebra cohomologies of $(\varphi ,\Gamma )$-modules. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 665-690. doi: 10.5802/jtnb.1336

[1] Laurent Berger Multivariable (φ,Γ)-modules and locally analytic vectors, Duke Math. J., Volume 165 (2016) no. 18, pp. 3567-3595 | MR | Zbl

[2] Laurent Berger; Peter Schneider; Bingyong Xie Rigid Character Groups, Lubin–Tate Theory, and (φ,Γ)-Modules, Memoirs of the American Mathematical Society, 263, American Mathematical Society, 2020 no. 1275, v+79 pages | Zbl

[3] Pierre Colmez Représentations localement analytiques de GL 2 (ℚ p ) et (φ,Γ)-modules, Represent. Theory, Volume 20 (2016) no. 9, pp. 187-248 | DOI | MR | Zbl

[4] David Eisenbud Commutative algebra with view Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer, 1995, xvi+785 pages | MR | Zbl

[5] Jean-marc Fontaine; Yi Ouyang Theory of p -adic Galois Representations, http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf

[6] Lionel Fourquaux; Bingyong Xie Triangulable 𝒪 F -analytic (φ q ,Γ)-modules of rank 2, Algebra Number Theory, Volume 7 (2013) no. 10, pp. 2545-2592 | MR | Zbl | DOI

[7] Jan Kohlhaase The cohomology of locally analytic representations, J. Reine Angew. Math., Volume 2011 (2011) no. 651, pp. 187-240 | MR | Zbl | DOI

[8] Benjamin Kupferer; Otmar Venjakob Herr-complexes in the Lubin–Tate setting, Mathematika, Volume 68 (2022) no. 1, pp. 74-147 | DOI | MR | Zbl

[9] Michel Lazard Groupes analytiques p-adiques, Publ. Math., Inst. Hautes Étud. Sci., Volume 26 (1965), pp. 5-219 | Numdam

[10] Ruochuan Liu Cohomology and duality for (φ, Γ)-modules over the Robba ring, Int. Math. Res. Not., Volume 2008 (2008), rnm150, 32 pages | MR | Zbl

[11] Milan Malcic; Rustam Steingart; Otmar Venjakob; Max Witzelsperger ϵ-isomorphisms for rank one (φ,Γ)-modules over Lubin-Tate Robba rings (2024) | arXiv | Zbl

[12] Barry Mitchell Theory of categories, Pure and Applied Mathematics, 17, Academic Press Inc., 1965, xi+273 pages | MR | Zbl

[13] Gal Porat Basic examples and computations in p-adic Hodge theory (unpublished notes)

[14] Peter Schneider Galois Representations and (φ,Γ)-Modules, Cambridge Studies in Advanced Mathematics, 164, Cambridge University Press, 2017, vii+148 pages | MR | DOI | Zbl

[15] Peter Schneider; Otmar Venjakob Compairing categories of Lubin–Tate (φ L ,Γ L )-modules, Results Math., Volume 78 (2023) no. 6, 230, 36 pages | MR | Zbl

[16] Peter Schneider; Otmar Venjakob Reciprocity laws for (φ L ,Γ L )-modules over Lubin–Tate extensions (2023) | arXiv

[17] The Stacks Project Authors The Stacks project, https://stacks.math.columbia.edu, 2021

[18] Rustam Steingart Finiteness of analytic cohomology of Lubin–Tate (φ L ,Γ L )-modules, J. Number Theory, Volume 263 (2024), pp. 24-78 | DOI | MR | Zbl

[19] Rustam Steingart Iwasawa cohomology of analytic (φ L ,Γ L )-modules, Acta Arith., Volume 216 (2024) no. 2, pp. 123-176 | DOI | MR | Zbl

[20] Charles A. Weibel An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1995, xiv+450 pages | Zbl

Cité par Sources :