Almost $\cal D$-split sequences and derived equivalences

TL;DR

This paper introduces almost D-split sequences and reveals a surprising link between derived equivalences and Auslander-Reiten sequences, expanding understanding of their interrelations in representation theory.

Contribution

It establishes a new connection between derived equivalences and Auslander-Reiten sequences through the concept of almost D-split sequences and BB-tilting modules.

Findings

Derived equivalences can be obtained from Auslander-Reiten sequences.

Introduces almost D-split sequences as a new concept.

Links Auslander-Reiten triangles with derived equivalences.

Abstract

In this paper, we introduce almost $\cal D$-split sequences and establish an elementary but somewhat surprising connection between derived equivalences and Auslander-Reiten sequences via BB-tilting modules. In particular, we obtain derived equivalences from Auslander-Reiten sequences (or $n$-almost split sequences), and Auslander-Reiten triangles.