Liftable derived equivalences and objective categories

TL;DR

This paper proves that derived equivalences involving algebras derived from smooth projective schemes are standard, using liftable functors and the triangle-objectivity of derived categories of coherent sheaves.

Contribution

It provides two proofs that all derived equivalences in this context are standard, extending the understanding of derived categories and equivalences in algebraic geometry.

Findings

Liftable functors coincide with standard functors between derived categories.

Derived equivalences between module and abelian categories factor uniquely into pseudo-identity and liftable equivalence.

Derived categories of coherent sheaves on certain schemes are triangle-objective, with autoequivalences being trivial.

Abstract

We give two proofs to the following theorem and its 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 projective scheme is triangle-objective, that is, any triangle autoequivalence on it, which preserves the the isomorphism classes of complexes, is necessarily isomorphic to the identity functor.