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 (K,δ) 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 (K,δ) of characteristic 0 is algebraically closed and closed under Picard–Vessiot extensions then every differential algebraic 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 : 10.5802/afst.1615
Chatzidakis, Zoé 1 ; Pillay, Anand 2

1 DMA - ENS, 45 rue d’Ulm, 75230 Paris cedex 05, France
2 Department of Mathematics, 255 Hurley, Notre Dame, IN 46556, USA
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Chatzidakis, Zoé; Pillay, Anand. 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. http://www.numdam.org/articles/10.5802/afst.1615/

[1] Cassidy, Phyllis J.; Singer, Michael F. 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 | Zbl

[2] Chatzidakis, Zoé; Hardouin, Charlotte; Singer, Michael F. 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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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