Hopf dense Galois extensions with applications

Let $H$ be a finite dimensional Hopf algebra, and let $A$ be a left $H$-module algebra. Motivated by the study of the isolated singularities of $A^H$ and the endomorphism ring $\mathrm{End}_{A^H}(A)$, we introduce the concept of Hopf dense Galois extensions in this paper. Hopf dense Galois extensions yield certain equivalences between the quotient categories over $A$ and $A^H$. A special class of Hopf dense Galois extensions consits 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, we recover the classical equivalence of the noncommutative projective schemes over a noetherian $\mathbb{N}$-graded algebra $A$ and its $d$-th Veroness subalgebra $A^{(d)}$ respectively. Hopf dense Galois extensions are also applied to the study of noncommuative graded isolated singularities.