Brauer group of the moduli spaces of stable vector bundles of fixed determinant over a smooth curve

Let X be an irreducible smooth projective curve, defined over an algebraically closed field k, of genus at least three and L a line bundle on X. Let MX(r,L) be the moduli space of stable vector bundles on X of rank r and determinant L with r≥2. We prove that the Brauer group Br(MX(r,L)) is cyclic of order g.c.d.(r,degree(L)). We also prove that Br(MX(r,L)) is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in [1] under the assumption that k=C.