Generalized Picard-Vessiot extensions and differential Galois cohomology

2017/02/07 by Zoé Chatzidakis, Anand Pillay, Chatzidakis, Zoe +1 · 1 citation
Mathematics · #12H05 03C60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1702.01969

openalex publication_date 2017/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In an earlier paper it was proved that if a differential field (K,δ) 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.

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• Trivializing Torsors for Difference Algebraic Groups

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