Geodesic
An extended form of the Grothendieck--Serre conjecture
Izvestiya. Mathematics , Tome 86 (2022) no. 4, pp. 782-796

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Let $R$ be a regular semi-local integral domain containing a field, $K$ the fraction field of $R$, and $\mu\colon \mathbf{G} \to \mathbf{T}$ an $R$-group scheme morphism between reductive $R$-group schemes which is smooth as a scheme morphism. Suppose that $\mathbf{T}$ is an $R$-torus. Then the map $\mathbf{T}(R)/ \mu(\mathbf{G}(R)) \to \mathbf{T}(K)/ \mu(\mathbf{G}(K))$ is injective and a purity theorem holds. These and other results can be derived from an extended form of the Grothendieck–Serre conjecture proven in the present paper for any such ring $R$.
Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture, purity theorem. 
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     author = {I. A. Panin},
     title = {An extended form of the {Grothendieck--Serre} conjecture},
     journal = {Izvestiya. Mathematics },
     pages = {782--796},
     publisher = {mathdoc},
     volume = {86},
     number = {4},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_4_a6/}
}
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I. A. Panin. An extended form of the Grothendieck--Serre conjecture. Izvestiya. Mathematics , Tome 86 (2022) no. 4, pp. 782-796. http://geodesic.mathdoc.fr/item/IM2_2022_86_4_a6/