Graded Morita equivalences for generic Artin-Schelter regular algebras

Abstract

Let $A=\mathcal{A}(E,\sigma ),A\text{'}=\mathcal{A}(E\text{'},\sigma \text{'})$ be Noetherian Artin-Schelter regular geometric algebras with ${dim}_k{A}_1={dim}_k{A}_1\text{'}=n$, and let $\nu ,\nu \text{'}$ be generalized Nakayama automorphisms of $A,A\text{'}$. In this paper, we study relationships between the conditions

(A) $A$ is graded Morita equivalent to $A\text{'}$, and

(B) $\mathcal{A}(E,{\nu }^{\ast }{\sigma }^n)$ is isomorphic to $\mathcal{A}(E\text{'},(\nu \text{'}{)}^{\ast }\left(\sigma \text{'}{)}^n\right)$ as graded algebras.

It is proved that if $A,A\text{'}$ are “generic” 3-dimensional quadratic Artin-Schelter regular algebras, then (A) is equivalent to (B), and if $A,A\text{'}$ are $n$-dimensional skew polynomial algebras, then (A) implies (B).