Generalized Picard–Vessiot extensions and differential Galois cohomology
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 813-830.

On a montré dans [18] que si un corps différentiel $\left(K,\delta \right)$ de caractéristique $0$ est algébriquement clos et clos par extensions de Picard–Vessiot, alors tout espace principal homogène différentiel algébrique sur $K$ a un point $K$-rationnel (et réciproquement). Cet article explore s’il est possible, et si oui comment, d’étendre ce résultat au cas de (a) plusieurs dérivations qui commutent, (b) un automorphisme. Pour une notion naturelle d’« extension de Picard–Vessiot généralisée » (dans le cas de plusieurs dérivations) nous donnons un contre-exemple. Nous avons aussi un contre-exemple dans le cas d’un automorphisme. Enfin, nous formulons et démontrons quelques résultats positifs dans le cas de plusieurs dérivations.

In [18] it was proved that if a differential field $\left(K,\delta \right)$ of characteristic $0$ is algebraically closed and closed under Picard–Vessiot extensions then every differential algebraic $\mathrm{PHS}$ over $K$ for a linear differential algebraic group $G$ over $K$ has a $K$-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several commuting derivations, (b) one automorphism. Under a natural notion of “generalized Picard–Vessiot extension” (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations.

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DOI : https://doi.org/10.5802/afst.1615
@article{AFST_2019_6_28_5_813_0,
author = {Zo\'e Chatzidakis and Anand Pillay},
title = {Generalized {Picard{\textendash}Vessiot} extensions and differential {Galois} cohomology},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {813--830},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 28},
number = {5},
year = {2019},
doi = {10.5802/afst.1615},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1615/}
}
Zoé Chatzidakis; Anand Pillay. Generalized Picard–Vessiot extensions and differential Galois cohomology. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 813-830. doi : 10.5802/afst.1615. https://afst.centre-mersenne.org/articles/10.5802/afst.1615/

[1] Phyllis J. Cassidy; Michael F. Singer Galois theory of parameterized differential equations and linear differential algebraic groups, Differential equations and quantum groups (IRMA Lectures in Mathematics and Theoretical Physics) Volume 9, European Mathematical Society, 2007, pp. 113-155 | MR 2322329 | Zbl 1356.12004

[2] Zoé Chatzidakis; Charlotte Hardouin; Michael F. Singer On the definitions of difference Galois groups, Model theory with applications to algebra and analysis. Vol. 1 (London Mathematical Society Lecture Note Series) Volume 349, Cambridge University Press, 2008, pp. 73-109 | Article | MR 2441376 | Zbl 1234.12005

[3] Zoé Chatzidakis; Ehud Hrushovski Model theory of difference fields, Trans. Am. Math. Soc., Volume 351 (1999) no. 8, pp. 2997-3071 | Article | MR 1652269 | Zbl 0922.03054

[4] Richard M. Cohn Difference algebra, Interscience Publishers, 1965 | Zbl 0127.26402

[5] Charlotte Hardouin; Michael F. Singer Differential Galois theory of linear difference equations, Math. Ann., Volume 342 (2008) no. 2, pp. 333-377 | Article | MR 2425146 | Zbl 1163.12002

[6] Moshe Kamensky Definable groups of partial automorphisms, Sel. Math., New Ser., Volume 15 (2009) no. 2, pp. 295-341 | Article | MR 2529938 | Zbl 1184.03028

[7] Ellis R. Kolchin Differential algebra and algebraic groups, Pure and Applied Mathematics, Volume 54, Academic Press Inc., 1973 | MR 568864 | Zbl 0264.12102

[8] Ellis R. Kolchin Differential algebraic groups, Pure and Applied Mathematics, Volume 114, Academic Press Inc., 1985 | MR 776230 | Zbl 0556.12006

[9] Piotr Kowalski; Anand Pillay On algebraic $\sigma$-groups, Trans. Am. Math. Soc., Volume 359 (2007) no. 3, pp. 1325-1337

[10] Peter Landesman Generalized differential Galois theory, Trans. Am. Math. Soc., Volume 360 (2008) no. 8, pp. 4441-4495 | Article | MR 2395180 | Zbl 1151.12004

[11] Omar León Sánchez; Anand Pillay Some definable Galois theory and examples, Bull. Symb. Log., Volume 23 (2017) no. 2, pp. 145-159 | Article | MR 3664720 | Zbl 1419.03032

[12] David Marker Differential fields, Model Theory of Fields (Lecture Notes in Logic) Volume 5, A K Peters; Association for Symbolic Logic, 2006 | Zbl 1104.12006

[13] Tracey McGrail The model theory of differential fields with finitely many commuting derivations, J. Symb. Log., Volume 65 (2000) no. 2, pp. 885-913 | Article | MR 1771092 | Zbl 0960.03031

[14] Andrei Minchenko; Alexey Ovchinnikov Triviality of differential Galois cohomologies of linear differential algebraic groups (https://arxiv.org/abs/1707.08620)

[15] Anand Pillay Geometric stability theory, Oxford Logic Guides, Volume 32, Clarendon Press, 1996 | MR 1429864 | Zbl 0871.03023

[16] Anand Pillay Some foundational questions concerning differential algebraic groups, Pac. J. Math., Volume 179 (1997) no. 1, pp. 179-200 | Article | MR 1452531 | Zbl 0999.12009

[17] Anand Pillay Differential Galois theory. I, Ill. J. Math., Volume 42 (1998) no. 4, pp. 678-699 | Article | MR 1649893 | Zbl 0916.03028

[18] Anand Pillay The Picard-Vessiot theory, constrained cohomology, and linear differential algebraic groups, J. Math. Pures Appl., Volume 108 (2017) no. 6, pp. 809-817 | Article | MR 3723157 | Zbl 1430.12009

[19] Jean-Pierre Serre Cohomologie Galoisienne, Lecture Notes in Mathematics, Volume 5, Springer, 1973 | MR 404227

[20] Sonat Süer On subgroups of the additive group in differentially closed fields, J. Symb. Log., Volume 77 (2012) no. 2, pp. 369-391 | Article | MR 2963012 | Zbl 1248.03054

[21] Katrin Tent; Martin Ziegler A course in model theory, Lecture Notes in Logic, Volume 40, Association for Symbolic Logic; Cambridge University Press, 2012 | MR 2908005 | Zbl 1245.03002

[22] André Weil On algebraic groups of transformations, Am. J. Math., Volume 77 (1955), pp. 355-391 | Article | MR 74083 | Zbl 0065.14201