Affineness on Noetherian graded rings, algebras and Hopf algebras
Huan Jia, Yinhuo Zhang
TL;DR
The paper investigates when Noetherian graded rings and graded Hopf algebras are affine, proving that if the degree-zero part $R_0$ (or $A_0$) is affine, then the entire graded ring or algebra is affine; this holds particularly when $A_0$ is a commutative or cocommutative Hopf subalgebra. It then analyzes the braided Hopf algebra arising in the Radford biproduct decomposition $H \cong B \sharp H_0$ of a Noetherian graded Hopf algebra, showing that the braided component $B$ is affine under Noetherian hypotheses. Consequently, for pointed Hopf algebras with $\operatorname{gr}_c H$ Noetherian, the coradical, the graded coradical, the algebra itself, and the braided part are all affine, with $\operatorname{gr}_c H \cong B \sharp \operatorname{corad}(H)$. These results advance understanding of when Noetherian graded Hopf algebras are affine and clarify the structure of their braided components, with implications for classification and structure theory in infinite-dimensional Hopf algebras.
Abstract
In this note, we show that every Noetherian graded ring with an affine degree zero part is affine. As a result, a Noetherian graded Hopf algebra whose degree zero component is a commutative or a cocommutative Hopf subalgebra is affine. Moreover, we show that the braided Hopf algebra of a Noetherian graded Hopf algebra is affine.