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On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:I | | I. Panin
; A. Stavrova
; N. Vavilov
; | | Date: |
9 May 2009 | | Abstract: | 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. | | Source: | arXiv, 0905.1418 | | Services: | Forum | Review | PDF | Favorites | |
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