Let and be schemes over a field . Let be an object in the derived category of quasi-coherent sheaves over their product. (This is a correspondence between and equipped with a chain complex of quasi-coherent sheaves).
The functor defined by
where and are the projections from onto and , respectively, is called the Fourier-Mukai transform of , or the Fourier-Mukai functor induced by .
When is isomorphic to for some , one also says that is represented by or simply that is of Fourier-Mukai type.
The key fact is as follows
See Orlov 2003, 3.2.1 for a proof. Though the theorem is stated there for admitting a right adjoint, it follows from Bondal-van den Bergh 2002 that every triangulated fully faithful functor admits a right adjoint automatically.
It is generally believed that this theorem should be true for all triangulated functors?.
On the level of the DG enhancements, it is true for all smooth proper -schemes that, in the homotopy category of DG categories, every functor corresponds bijectively to an isomorphism class of objects on . See (Toen 2006).
Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv)
Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.
Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667
Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)
For a discussion of Fourier-Mukai transforms in the setting of -enhancements, see