Representability of cohomological functors over extension fields

We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor H : D-Coh(b)(X) -> (mod)under bar(L) to the case where L is a field extension of the base field k of the variety X, with trdeg(k)L <= 1 or L purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor F : D-Coh(b)(X) -> D-Coh(b)(Y) with X and Y smooth projective varieties and dim Y <= 1 or Y a rational surface. We show that for any such F there exists a "generic kernel" A in D-Coh(b)(X x Y), 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.