Fixed Subrings of Noetherian Graded Regular Rings

Rings of invariants can have nice homological properties even if they do not have finite global dimension. Watanabe’s Theorem [W] gives conditions when the fixed subring of a commutative ring under the action of a finite group is a Gorenstein ring. The Gorenstein condition was extended to noncommutative rings by a condition explored by Idun Reiten in the 1970s, called k-Gorenstein in [FGR]. This condition, also known as the Auslander-Gorenstein condition, has proved to be a very useful one, and now has been generalized further in the notion of an Auslander dualizing complex (see e.g. [YZ]). Artin and Schelter defined another Gorenstein condition for connected graded rings (see [AS]); this condition is now called the Artin-Schelter Gorenstein condition. Noncommutative versions of Watanabe’s Theorem, giving conditions when a fixed ring satisfies a Gorenstein condition, were proved by P. Jørgensen and J. Zhang [JoZ], and extended by N. Jing and J. Zhang [JZ2]. These conditions involve the “homological determinant” of the automorphisms in the group, as defined in [JoZ]. In this paper we apply these results to down-up algebras, their extensions, and certain generalized Weyl algebras, and thus expand the class of algebras for which the homological determinant can be easily computed. The fact that rings of invariants in some cases give algebras that are Artin-Schelter regular was one motivation for this paper. The homological determinant defines a group homomorphism from the group of graded automorphisms of a connected graded K-algebra A to the multiplicative group of the field K∗; its name is due to the fact that when A = K[x1, . . . , xn] is a commutative polynomial ring each graded automorphism g is associated to an