Generally, for two suitably well-behaved schemes (e.g. affine, smooth, complex) and with , their derived categories of quasicoherent sheaves, then a Fourier-Mukai transform with integral kernel is a functor (of triangulated categories/stable (infinity,1)-categories)
which is given as the composite of the (derived) operations of
tensor product with ;
push (direct image) along the other projection
with twist on the correspondence space.
As discussed at integral transforms on sheaves this kind of integral transform is a categorification of an integral transform/matrix multiplication of functions induced by an integral kernel, the role of which here is played by .
Indeed, the central kind of result of the theory (theorem 1) says that every suitable linear functor arises as a Fourier-Mukai transform for some , a statement which is the categorification of the standard fact from linear algebra that every linear function between finite dimensional vector spaces is represented by a matrix.
If is a moduli space of line bundles over a suitable algebraic curve, then a slight variant of the Fourier-Mukai transform is the geometric Langlands correspondence in the abelian case (Frenkel 05, section 4.4, 4.5).
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 theorem 1 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 (see e.g. Huybrechts 08, p. 6).
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).
Shigeru Mukai, Duality between and with its application to Picard sheaves. Nagoya Mathematical Journal 81: 153–175. (1981)
Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv:0402043)
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)
Discussion in the context of geometric Langlands duality is in
For a discussion of Fourier-Mukai transforms in the setting of -enhancements, see