A short exact sequence

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.