[Paper Review] Auslander's Theorem for group coactions on noetherian graded down-up algebras

Jiangxi Chen, Ellen Kirkman+1 more|arXiv (Cornell University)|Jan 27, 2018

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper establishes a noncommutative version of Auslander's Theorem for finite group coactions on noetherian graded down-up algebras. By proving that the pertinency is at least 2 under the condition of trivial homological determinant, it confirms the isomorphism between the smash product algebra $A\#H$ and the endomorphism algebra $\operatorname{End}_{A^H}(A)$, thereby extending the noncommutative McKay correspondence to this class of algebras.

ABSTRACT

We prove a version of a theorem of Auslander for finite group coactions on noetherian graded down-up algebras.

Motivation & Objective

• To extend Auslander's Theorem to the setting of finite group coactions on noetherian graded down-up algebras.
• To verify the noncommutative Auslander conjecture for Artin-Schelter regular down-up algebras of global dimension three.
• To establish conditions under which the natural map $A\#H \to \operatorname{End}_{A^H}(A)$ is an isomorphism.
• To demonstrate that the pertinency condition $\mathsf{p}(A,H) \geq 2$ holds when the homological determinant is trivial.

Proposed method

• Use the pertinency invariant $\mathsf{p}(A,H) = \operatorname{GKdim} A - \operatorname{GKdim}((A\#H)/I)$, where $I$ is the ideal generated by $1\#\int$, to analyze the action of the Hopf algebra $H = \Bbbk^G$.
• Apply the equivalence between left $H$-actions and right $K$-coactions, with $K = \Bbb{k}G$ for a finite group $G$, to reframe the problem in terms of coactions.
• Leverage the fact that $A = \mathbb{D}(\alpha,\beta)$ is Artin-Schelter regular of global dimension three and noetherian when $\beta \neq 0$, ensuring suitable homological properties.
• Use the triviality of the homological determinant as a key condition to ensure $\mathsf{p}(A,H) \geq 2$, which implies the isomorphism $A\#H \cong \operatorname{End}_{A^H}(A)$ via [4, Theorem 0.3].
• Verify the isomorphism by checking that the canonical map $\phi: A\#H \to \operatorname{End}_{A^H}(A)$ is an isomorphism of graded algebras under the given conditions.
• Analyze specific examples, such as $D_{2n}$-coactions on the algebra $\mathbb{H}$, to show that the fixed subalgebra is a hypersurface in an iterated Ore extension, confirming the structure of the invariant ring.

Experimental results

Research questions

• RQ1 Does Auslander's Theorem hold for finite group coactions on noetherian graded down-up algebras when the homological determinant is trivial?
• RQ2 Under what conditions is the pertinency $\mathsf{p}(A,H)$ at least 2 for such coactions?
• RQ3 Can the isomorphism $A\#H \cong \operatorname{End}_{A^H}(A)$ be established in the noncommutative setting of down-up algebras?
• RQ4 Is the noncommutative McKay correspondence realizable for down-up algebras via this isomorphism?
• RQ5 Does the failure of the theorem occur when the homological determinant is nontrivial, as shown in Remark 1.6(2)?

Key findings

• The paper proves that for a finite group $G$ acting on the noetherian graded down-up algebra $\mathbb{D}(\alpha,\beta)$ via $H = \Bbb{k}^G$, if the homological determinant is trivial, then $\mathsf{p}(A,H) \geq 2$.
• This implies the existence of a graded algebra isomorphism $A\#H \cong \operatorname{End}_{A^H}(A)$, confirming the noncommutative Auslander theorem in this setting.
• The result holds specifically when $\beta \neq 0$, ensuring the algebra is noetherian and Artin-Schelter regular of global dimension three.
• The theorem fails without the trivial homological determinant condition, as demonstrated in Remark 1.6(2).
• The fixed subalgebra $\mathbb{H}^{co\;D_{2n}}$ is shown to be a hypersurface in an iterated Ore extension, with Hilbert series $\frac{1-t^{4n}}{(1-t^2)^2(1-t^{2n})^2}$, confirming its AS-regular structure.
• The construction of the invariant ring via generators $x^2, y^2, (yx)^n, (xy)^n$ and relations involving $\sigma$-twisted derivations confirms the algebra is a generalized Weyl algebra.