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

Abstract

Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme isotropic over R. We prove that for any Noetherian k-algebra A, the map of etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced by the inclusion of R into K, has trivial kernel. This implies the Serre-Grothendieck conjecture for such groups G. The main theorem for A=k and some other results of the present paper are used significantly in arXiv:1211.2678 to prove the Serre-Grothendieck conjecture for all reductive groups over a regular semi-local ring containing an infinite field.

We have incorporated arXiv:1204.1729 and arXiv:0910.5465 into this text

References in corpus (1)

• On the norm principle for quadratic forms

Cited by in corpus (7)

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• Grothendieck-Serre Conjecture I: Appendix
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• Grothendieck-Serre conjecture for groups of type F_4 with trivial f_3 invariant
• Nice triples and a moving lemma for motivic spaces
• R-equivalence on group schemes
• Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles