Auslander's Theorem Research Articles

Actions of small groups on two-dimensional Artin-Schelter regular algebras

In commutative invariant theory, a classical result due to Auslander says that if R=k[x1,…,xn] and G is a finite subgroup of Autgr(R)≅GL(n,k) which contains no reflections, then there is a natural graded isomorphism R#G≅EndRG(R). In this paper, we show that a version of Auslander's Theorem holds if we replace R by an Artin-Schelter regular algebra A of global dimension 2, and G by a finite subgroup of Autgr(A) which contains no quasi-reflections. This extends work of Chan–Kirkman–Walton–Zhang. As part of the proof, we classify all such pairs (A,G), up to conjugation of G by an element of Autgr(A). In all but one case, we also write down explicit presentations for the invariant rings AG, and show that they are isomorphic to factors of AS regular algebras.

Fixed rings of generalized Weyl algebras

We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical GWA, extending a result of Jordan and Wells. Partial results are provided for the case of higher degree polynomials. In addition, we establish a version of Auslander's theorem for finite cyclic groups of filtered automorphisms acting on classical GWAs.

Auslander’s Theorem for permutation actions on noncommutative algebras

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.

McKay correspondence for semisimple Hopf actions on regular graded algebras, I

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin–Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring AH under such an action arises as an analogue of a coordinate ring of a Kleinian singularity.

Gorenstein Analogue of Auslander's Theorem on the Global Dimension

In this paper, we prove that the Gorenstein analogue of the well-known Auslander's theorem on the global dimension holds true. Namely, we prove that the Gorenstein global dimension of a commutative ring R is equal to the supremum of the set of Gorenstein projective dimensions of all cyclic R-modules.

An Auslander-type result for Gorenstein-projective modules

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105–179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269–310].

Auslander's theorem on discontinuous semigroups and crystallographic semigroups

(1996). Auslander's theorem on discontinuous semigroups and crystallographic semigroups. Communications in Algebra: Vol. 24, No. 4, pp. 1435-1444.