Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities

Abstract

In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal{D}}^b(\mathrm{tails}{S}^G)\cong {\mathcal{D}}^b(\mathrm{mod}\nabla S\ast G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S,G}$ such that ${\mathcal{D}}^b(\mathrm{tails}{S}^G)\cong {\mathcal{D}}^b(\mathrm{mod}k{Q}_{S,G})$ when $S$ is of dimension 2.

25 pages