2009/05/09 by I. Panin, Panin, I., A. Stavrova +3 · 4 citations
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #math.AG #math.GR
paper · pdf · doi:10.48550/arxiv.0905.1418
We have incorporated arXiv:1204.1729 and arXiv:0910.5465 into this text
arxiv created 2013/04/25 · arxiv updated 2013/04/26
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 H1(A⊗k R,G)-> H1(A⊗_ 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.