Affine commutative-by-finite Hopf algebras

The objects of study in this paper are Hopf algebras H which finitely generated over an algebraically closed field and extensions a commutative algebra A by finite dimensional H?:=H/A+H, where A+ is the augmentation ideal A. Basic structural homological properties recalled classes examples listed. structure theorem proved when (?) H? semisimple cosemisimple, showing that case noncommutativity arises from action group. For example, holds prime pointed, it crossed product smooth affine domain group, simple H-modules described type Clifford's theorem.