Let E be a principal G–bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

Let G be an affine reductive algebraic group over an algebraically closed field k . We determine the Picard group of the moduli stacks of principal G -bundles on any smooth projective curve over k .