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Universal derived equivalences of posets of cluster tilting objects
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We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.
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Derived equivalence of posets of torsion classes
Algebras related by 1-APR tilt have torsion class posets related by flip-flop, so their incidence algebras are derived equivalent.
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