Let F be the function field of a curve over a complete discretely valued field K. Let G be a semisimple simply connected linear algebraic group over F of type \(A_n\) . We give a description of the obstruction to the local global principle for principal homogeneous spaces under G over F with respect to discrete valuations of F in terms of R-equivalence classes of G over some suitable over fields. Using this description we prove that this obstruction vanishes when the residue field of K has cohomological dimension at most 1 and the characteristic exponent is prime to \(2(n+1)\) . We also prove the vanishing of this obstruction if G is defined over K and the residue field of K has cohomological dimension at most 2.
