An extended form of the Grothendieck–Serre conjecture

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 $R$.