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

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

2009-05-09

arxiv.org/abs/0905.1418v1

Mathematics

Algebraic Geometry

28 pages

Scientific paper

Let R be a semi-local regular domain containing an infinite perfect subfield k and let K be its field of fractions. 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 in this case the kernel of the map H^1_{et}(R,G) -> H^1_{et}(K,G) induced by the inclusion of R into K is trivial. In other words, under the above assumptions every principal G-bundle P which has a K-rational point is itself trivial. This confirms a conjecture posed by Serre and Grothendieck. Our proof is based on a combination of methods of Raghunathan's paper "Principal bundles admitting a rational section", Ojanguren--Panin's paper "Rationally trivial hermitian spaces are locally trivial", and Panin's preprint "A purity theorem for linear algebraic groups" (www.math.uiuc.edu/K-theory/0729). 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.

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