Trivial extensions of Koszul Artin-Schelter regular algebras

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 $S\ltimes S_\sigma(-1)$. 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.