Trivial extensions of Koszul Artin-Schelter regular algebras

TL;DR

This paper explores the stable category of Cohen-Macaulay modules over trivial extensions of Koszul Artin-Schelter regular algebras, revealing their equivalence to derived categories of modules over dual algebras.

Contribution

It establishes a triangle equivalence between the stable category of Cohen-Macaulay modules over trivial extensions and derived categories of modules over the Koszul dual, providing criteria for equivalences.

Findings

Stable category of Cohen-Macaulay modules is equivalent to derived category of modules over the dual algebra.

Criteria are provided for when two such categories are triangle equivalent.

Equivalence induces an isomorphism between categories of graded modules over the original 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 $S\ltimes S_\sigma(-1)$. 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.