McKay-type correspondence for AS-regular algebras

Abstract

The classical McKay correspondence claims that the minimal resolution of Spec k[x, y]G where G ⩽ SL(2, k) is a finite subgroup naturally acting on k[x, y] is derived equivalent to the preprojective algebra of the McKay quiver of G. In this paper, we will see that there is a similar correspondence in the case of a finite cyclic subgroup G ⩽ GL(n, k) naturally acting on an AS-regular algebra in n variables. Since AS-regular algebras are non-commutative analogues of polynomial algebras, this can be thought of as a McKay-type correspondence in non-commutative algebraic geometry.