Geodesic
A short exact sequence
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 176-186

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Let $R$ be a semi-local integral Dedekind domain and $K$ be its fraction field. Let $\mu: \mathbf{G} \to \mathbf{T}$ be an $R$-group schemes morphism between reductive $R$-group schemes, which is smooth as a scheme morphism. Suppose that $T$ 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 certain purity theorem is true. These and other results are derived from an extended form of Grothendieck–Serre conjecture proven in the present paper for rings $R$ as above. 
@article{ZNSL_2019_485_a9,
     author = {I. Panin},
     title = {A short exact sequence},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {176--186},
     publisher = {mathdoc},
     volume = {485},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a9/}
}
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I. Panin. A short exact sequence. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 176-186. http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a9/