  We generalize a result of Orlov and Van den Bergh on the representability of
a cohomological functor from the bounded derived category of a smooth
projective variety over a field to the category of L-modules, to the case where
L is a field extension of the base field k of the variety X, with L of
transcendence degree less than or equal to one or L purely transcendental of
degree 2.
  This result can be applied to investigate the behavior of an exact functor
between the bounded derived categories of coherent sheaves of X and Y, with X
and Y smooth projective and Y of dimension less than or equal to one or Y a
rational surface. We show that for any such F there exists a "generic kernel" A
in the derived category of the product, such that F is isomorphic to the
Fourier-Mukai transform with kernel A after composing both with the pullback to
the generic point of Y.

Algebraic Geometry