For coprime positive a,b, the derived category of the product of a type-A path algebra with the rational Dyck poset is equivalent to the derived category of the full path lattice, proving the CLR conjecture.
Derived equivalence of posets of torsion classes
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abstract
We investigate the structure of the poset of torsion classes for a finite dimensional algebra admitting a simple projective module. We generalise a result of Ladkani by showing that if two algebras are related by a 1-APR tilt, then their posets of torsion classes are related by a flip-flop. This implies that the incidence algebras of the posets are derived equivalent. We give two different proofs of this result. The first one applies to functorially finite torsion classes, and a key ingredient is to see the two posets we want to relate as subposets of a common poset of silting objects. The second proof applies to arbitrary torsion classes, and we use a similar strategy, this time embedding the two posets into a common poset of s-torsion pairs.
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Relative Interval Tilting, Higher Auslander Staircase Corners and Rational Dyck Posets
For coprime positive a,b, the derived category of the product of a type-A path algebra with the rational Dyck poset is equivalent to the derived category of the full path lattice, proving the CLR conjecture.