Mathematics – Algebraic Geometry
Scientific paper
2009-05-09
Mathematics
Algebraic Geometry
38 pages
Scientific paper
Let R be a regular semi-local ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive R-group scheme satifying a mild "isotropy condition". Then each principal G-bundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Two main Theorems of Panin's, Stavrova's and Vavilov's paper "On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I" state the same results for semi-simple simply connected R-group schemes. Our proof is heavily based on these results, and on two purity Theorems proven in the present preprint. One of these purity results looks as follows. Given an R-torus C and a smooth Given a smooth R-group scheme morphism $\mu: G \to C$ of reductive R-group schemes, with a torus C one can form a functor from R-algebras to abelian groups $S \mapsto {\cal F}(S):=C(S)/\mu(G(S))$. Assuming additionally that the kernel $ker(\mu)$ of $\mu$ is a reductive R-group scheme, we prove that this functor satisfies a purity theorem for R. If R is of geometric type, then it suffices to assume that R is of geometric type over an arbitrary infinite field. Examples to mentioned purity results are considered in the very end of the preprint.
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