An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves

TL;DR

This paper demonstrates that Orlov's theorem, which characterizes certain functors between derived categories as Fourier-Mukai, does not hold if the functor is not fully faithful, highlighting limitations of the original result.

Contribution

The paper provides a counterexample showing that non-Fourier-Mukai functors can exist between derived categories when full faithfulness is absent.

Findings

Counterexample to Orlov's theorem without full faithfulness

Full faithfulness is essential for Fourier-Mukai characterization

Limits of existing representability results for derived functors

Abstract

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.