An example of a non-Fourier-Mukai functor between derived categories of   coherent sheaves
  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. 
Alice Rizzardo, Michel Van den Bergh, Amnon Neeman
math.AG