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

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1615
Zoé Chatzidakis 1; Anand Pillay 2

1 DMA - ENS, 45 rue d’Ulm, 75230 Paris cedex 05, France
2 Department of Mathematics, 255 Hurley, Notre Dame, IN 46556, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Zoé Chatzidakis; Anand Pillay. Generalized Picard–Vessiot extensions and differential Galois cohomology. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 813-830. doi : 10.5802/afst.1615. https://afst.centre-mersenne.org/articles/10.5802/afst.1615/

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