Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

TL;DR

This paper proves that in a specific 2-Calabi--Yau setting, maximal rigid objects with self-injective endomorphism algebras have derived equivalent endomorphism algebras, and describes the tilting complex inducing this equivalence.

Contribution

It establishes derived equivalences between endomorphism algebras of maximal rigid objects in 2-Calabi--Yau categories and explicitly describes the tilting complex involved.

Findings

Endomorphism algebras are derived equivalent.

Explicit description of the tilting complex.

Applicable to self-injective 2-Calabi--Yau tilted algebras.

Abstract

Consider a $k$-linear Frobenius category $\mathscr{E}$ with a projective generator such that the corresponding stable category $\mathscr{C}$ is 2-Calabi--Yau, Hom-finite with split idempotents. Let $l,m\in\mathscr{C}$ be maximal rigid objects with self-injective endomorphism algebras. We will show that their endomorphism algebras $\mathscr{C}(l,l)$ and $\mathscr{C}(m,m)$ are derived equivalent. Furthermore we will give a description of the two-sided tilting complex which induces this derived equivalence.