Generalized Picard–Vessiot extensions and differential Galois cohomology

Abstract

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://afst. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Annales de la faculté des sciences de Toulouse Volume XXVIII, n o 5, 2019 pp. 813-830 Generalized Picard-Vessiot extensions and differential Galois cohomology (*) Zoé Chatzidakis (1) and Anand Pillay (2) ABSTRACT.-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. RÉSUMÉ.-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.

Difference sheaves and torsors

“…In [7,Proposition 5.6], there is an example of a strongly PV-closed difference field with non-trivial linear difference torsors (such a situation cannot happen in the differential case; see [21]). Using Corollary 7.7 and Remark 7.13, one can interpret this result in terms of difference cohomology in the following way.…”

“…Proof. In the proof of [7,Proposition 5.6], there is a construction of a strongly PV-closed difference field (k, s) and a difference torsor of (G m , x → x 2 ) which has no difference (k, s)-rational points. By Remark 7.4(2), this difference torsor is not trivial, hence we get the result by Corollary 7.7 (and Definition 7.9).…”

“…In this subsection, we analyze two well-known constructions and interpret some results from [7] in terms of difference Galois cohomology. Let G be a group scheme over X.…”

“…We will use the second construction above to interpret in terms of difference cohomology some results about difference Picard-Vessiot extensions from [7]. A difference field (k, s) is called strongly PV-closed [7, Definition 5.1] if all linear difference equations over k have solutions in (k, s) (with the extra assumption of k being algebraically closed, this notion goes by the name of linearly closed in [16,Definition 5.1]).…”

Triviality of differential Galois cohomology of linear differential algebraic groups

“…Such a question for m = 1 was settled in [14], in which case the "linearly differentially closed" condition does not play a role. This was extended to m > 1 in [5] in terms of generalized strongly normal extensions. Our characterization is different and our arguments can be viewed as more transparent.…”