A construction of derived equivalent pairs of symmetric algebras

TL;DR

This paper broadens the class of triangles in algebraic triangulated categories that produce derived equivalent endomorphism rings, enabling the construction of new pairs of derived equivalent symmetric algebras with applications in various algebraic contexts.

Contribution

It introduces a general framework for deriving equivalences from a wider class of triangles in algebraic triangulated categories, extending previous results by Hu and Xi.

Findings

Constructed new pairs of derived equivalent symmetric algebras.

Applied results to categories of perfect complexes over symmetric algebras.

Provided examples involving dihedral type algebras and Gorenstein hypersurfaces.

Abstract

Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated categories) for which the endomorphism rings of different terms are derived equivalent. We then study applications involving 0-Calabi-Yau triangulated categories. In particular, applying our results in the category of perfect complexes over a symmetric algebra gives a nice way of producing pairs of derived equivalent symmetric algebras. Included in the examples we work out are some of the algebras of dihedral type with two or three simple modules. We also apply our results to stable categories of Cohen-Macaulay modules over odd-dimensional Gorenstein hypersurfaces having an isolated singularity.