Trivial extensions of Koszul Artin-Schelter regular algebras

Abstract

Let $S$ be an $\mathbb{N}$-graded Koszul Artin-Schelter regular algebra and let $\sigma$ be a graded algebra automorphism of $S$. We study the stable category of graded maximal Cohen-Macaulay modules over the trivial extension algebra $S⋉S_{\sigma }(-1)$. We show that this category is triangle equivalent to the bounded derived category of finitely generated (ungraded) modules over the Koszul dual algebra of the Zhang twist $S^{{\sigma }^{-1}}$. In the connected graded case, we also obtain a criterion for when two such stable categories are triangle equivalent, and show that such an equivalence induces an equivalence between the categories of graded modules over the original algebras.

19 pages