Hopf dense Galois extensions with applications (Q511850)

The study of Hopf algebra actions on Artin-Schelter regular algebras is an important subject in the field of noncommutative algebraic geometry. Many interesting results have been obtained in recent years. Let \(H\) be a finite dimensional Hopf algebra, and \(A\) a left \(H\)-module algebra. Motivated by the study of the isolated singularities of \(A^H\) and the endomorphism ring \(\text{End}_{A^H}(A)\), the aim of the paper under review is to develop a general theory on Hopf dense Galois extensions, and then in this framework to understand the finite dimensional Hopf-actions on Artin-Schelter regular algebras. Hopf dense Galois extensions yield certain equivalences between the quotient categories over \(A\) and \(A^H\). A special class of Hopf dense Galois extensions consists of the so-called densely group graded algebras, which are weaker versions of strongly graded algebras. A weaker version of Dade's theorem holds for densely group graded algebras. As applications, the authors recover the classical equivalence of the noncommutative projective scheme over a Noetherian \(\mathbb{N}\)-graded algebra \(A\) and its \(d\)-th Veronese subalgebra \(A^{(d)}\) respectively. Hopf dense Galois extensions are also applied to the study of noncommutative graded isolated singularities.