Affineness on Noetherian graded algebras and graded Hopf algebras

In this note, we show that a right or left Noetherian graded algebra is affine if and only if its degree-zero part is affine. This result implies that a Noetherian Hopf algebra, when graded as an algebra, is affine if its degree-zero component is either a commutative or a cocommutative Hopf subalgebra. We further establish that the braided Hopf algebra of any right or left Noetherian graded Hopf algebra is affine. Communicated by Eric Jespers