Generalized Picard-Vessiot extensions and differential Galois cohomology

Zoe Chatzidakis; Anand Pillay

Generalized Picard-Vessiot extensions and differential Galois cohomology

Algebraic Geometry 2017-09-12 v2

Authors: Zoe Chatzidakis , Anand Pillay

Abstract

In an earlier paper it was proved that if a differential field $(K,\delta)$ is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic principal homogeneous space 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 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.

Keywords

differential field Picard-Vessiot extension differential Galois cohomology principal homogeneous space linear differential algebraic group derivations automorphism

Cite

@article{arxiv.1702.01969,
  title  = {Generalized Picard-Vessiot extensions and differential Galois cohomology},
  author = {Zoe Chatzidakis and Anand Pillay},
  journal= {arXiv preprint arXiv:1702.01969},
  year   = {2017}
}

Comments

The new version is the one accepted by Annales Math. Toulouse, and makes some clarifications and additions suggested by the referees, including reference to a recent related paper by Minchenko and Ovchinnikov

Generalized Picard-Vessiot extensions and differential Galois cohomology

Abstract

Keywords

Cite

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