Geodesic
On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:~II
Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 759-790

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A proof of the Grothendieck–Serre conjecture on principal bundles over a semi-local regular ring containing an infinite field is given in [1]. That proof is heavily based on Theorem 1.0.3 stated below in the introduction and proved in the present paper. Theorem 1.0.3 itself is a consequence of two purity theorems 1.0.1 and 1.0.2 which are of completely independent interest and which are proved below. The purity theorem 1.0.1 covers all the known results of this shape and looks like a final one.
Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture. 
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     author = {I. A. Panin},
     title = {On {Grothendieck--Serre's} conjecture concerning principal $G$-bundles over reductive group {schemes:~II}},
     journal = {Izvestiya. Mathematics },
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     number = {4},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_4_a7/}
}
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I. A. Panin. On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:~II. Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 759-790. http://geodesic.mathdoc.fr/item/IM2_2016_80_4_a7/