  Let R be a regular semi-local integral domain containing a field and K be its
fraction field. Let mu: G --> 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 T(R)/mu(G(R)) --> T(K)/mu(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.
