Principal bundles of reductive groups over affine schemes

2012/04/08 by Ivan Panin, Anastasia Stavrova, Panin, Ivan +1
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1204.1729

This text has been incorporated into arXiv:0905.1418

openalex publication_date 2012/04/08 · arxiv created 2013/03/16 · arxiv updated 2013/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is isotropic. We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H¹(A×_k R,G)-> H¹(A×_ k K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local ring of several points on a k-smooth scheme, then it suffices to require that k is infinite and keep the same assumption concerning G.

Citations

• On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I

Cited by

• On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I

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