\(\mathcal D\)-split sequences and derived equivalences. (English) Zbl 1260.16017
Summary: We introduce the so-called \(\mathcal D\)-split sequences and show that each \(\mathcal D\)-split sequence gives rise to a derived equivalence via a tilting module. In particular, we obtain derived equivalences from Auslander-Reiten sequences via BB-tilting modules (or from \(n\)-almost split sequences via \(n\)-BB-tilting modules), and Auslander-Reiten triangles. Further, we recover \(n\)-almost split sequences from \(n\)-BB-tilting modules over \(n\)-Auslander algebras.
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G10 | Representations of associative Artinian rings |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16E35 | Derived categories and associative algebras |
Keywords:
\(\mathcal D\)-split sequences; Auslander-Reiten triangles; BB-tilting modules; derived equivalences; stable equivalences; Auslander-Reiten sequences; almost split sequencesReferences:
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