Liftable derived equivalences and objective categories

Abstract

We give two proofs of the following theorem and a partial generalization: if a finite-dimensional algebra A is derived equivalent to a smooth projective scheme, then any derived equivalence between A and another algebra B is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain class of projective schemes is triangle-objective, that is, any triangle autoequivalence on it, which preserves the isomorphism classes of all objects, is necessarily isomorphic to the identity functor.

Singular equivalences induced by bimodules and quadratic monomial algebras

“…The above triangle functor X ⊗ L B − : D sg (B) → D sg (A) clearly lifts to a morphism S dg (B) → S dg (A) in Hodgcat, the homotopy category of small dg categories [21]. In view of the derived Morita theory [22,6], the following two questions seem to be fundamental: does any triangle functor D sg (B) → D sg (A) lift to Hodgcat? How to characterize the morphism set in Hodgcat between dg singularity categories?…”