A Noetherian Hopf algebra is affine iff its Hopf coradical is affine

Abstract

Let $\K$ denote a field. Extending the structural frameworks established in \cite{JZ2025-2}, this paper introduces novel techniques utilizing non-commutative reduction orders, factorization theory, and the generalized lifting methodology. We establish a definitive necessary and sufficient criterion for the affineness of Noetherian Hopf algebras, thereby providing a significant advancement toward resolving the long-standing Wu--Zhang question \cite{WZ2003}. Specifically, we prove that a left or right Noetherian Hopf algebra over $\K$ is affine if and only if its Hopf coradical is affine. This characterization fundamentally concentrates the burden of verification onto the first filtration step, yielding a criterion that is structurally transparent and highly operational. To establish necessity, we provide an essential intrinsic result demonstrating that the Hopf coradical of an affine Hopf algebra inherits the property of being affine. Furthermore, as direct applications of this equivalence, we prove that a left or right Noetherian Hopf algebra $H$ is affine provided that its coradical $H_{(0)}$ forms a subalgebra (the dual Chevalley property), its coradical $H_{(0)}$ is cocommutative, or its Hopf coradical $H_{[0]}$ is commutative.