Hopf dense Galois extensions with applications

TL;DR

This paper introduces Hopf dense Galois extensions, a new concept linking Hopf algebra actions to category equivalences, with applications to noncommutative singularities and projective schemes.

Contribution

It defines Hopf dense Galois extensions, extends Dade's theorem for densely group graded algebras, and applies these ideas to noncommutative geometry and singularity theory.

Findings

Establishes equivalences between quotient categories over $A$ and $A^H$

Introduces densely group graded algebras as a weaker alternative to strongly graded algebras

Applies the theory to noncommutative projective schemes and isolated singularities

Abstract

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.