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The Fourier-Mukai transform is a categorified integral transform roughly similar to the standard Fourier transform.
Generally, for $X,Y$ two suitably well-behaved schemes (e.g. affine, smooth, complex) and with $D(X)$, $D(Y)$ their derived categories of quasicoherent sheaves, then a Fourier-Mukai transform with integral kernel $E \in D(X\times Y)$ is a functor (of triangulated categories/stable (infinity,1)-categories)
which is given as the composite of the (derived) operations of
pull (inverse image) along the projection $p_X\colon X\times Y \to X$
tensor product with $E$;
push (direct image) along the other projection $p_Y \colon X\times Y \to Y$
i.e.
(where here we implicitly understand all operations as derived functors). (e.g. Huybrechts 08, page 4)
Hence this is a pull-tensor-push integral transform through the product correspondence
with twist $E$ on the correspondence space.
Such concept of integral transform is rather general and may be considered also in derived algebraic geometry (e.g. BenZvi-Nadler-Preygel 13) and lots of other contexts.
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 $E\in D(X \times Y)$.
Indeed, the central kind of result of the theory (theorem 1) says that every suitable linear functor $D(X)\to D(Y)$ arises as a Fourier-Mukai transform for some $E$, 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.
The original Fourier-Mukai transform proper is the special case of the above where $X$ is an abelian variety, $Y = A^\vee$ its dual abelian variety and $E$ is the corresponding Poincaré line bundle.
If $X$ 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 $X$ and $Y$ be schemes over a field $K$. Let $E \in D(QCoh(O_{X \times Y}))$ be an object in the derived category of quasi-coherent sheaves over their product. (This is a correspondence between $X$ and $Y$ equipped with a chain complex $E$ of quasi-coherent sheaves).
The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by
where $p$ and $q$ are the projections from $X \times Y$ onto $X$ and $Y$, respectively, is called the Fourier-Mukai transform of $E$, or the Fourier-Mukai functor induced by $E$.
When $F : D(QCoh(O_X)) \to D(QCoh(O_Y))$ is isomorphic to $\Phi(E)$ for some $E \in D(QCoh(O_{X \times Y}))$, one also says that $F$ is represented by $E$ or simply that $F$ is of Fourier-Mukai type.
The key fact is as follows
Let $X$ and $Y$ be smooth projective varieties over a field $K$. Let $F : D(X) \to D(Y)$ be a triangulated fully faithful functor. Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.
See Orlov 2003, 3.2.1 for a proof.
Though theorem 1 is stated there for $F$ 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).
It was believed that theorem 1 should be true for all triangulated functors (e.g. Huybrechts 08, p. 5). However according to (RVdB 2015) this is not true.
On the level of the DG enhancements, it is true for all smooth proper $K$-schemes that, in the homotopy category of DG categories, every functor corresponds bijectively to an isomorphism class of objects on $D(X \times Y)$. See (Toen 2006).
Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves. Nagoya Mathematical Journal 81: 153–175. (1981)
Alexei Bondal, Michel van den Bergh. Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv
Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172, translation.
Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv:0402043)
Daniel Huybrechts, Fourier-Mukai transforms, 2008 (pdf)
Alice Rizzardo, Michel Van den Bergh, An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves (arXiv:1410.4039)
Pieter Belmans, section 2.2 of Grothendieck duality: lecture 3, 2014 (pdf)
Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.
Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in
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 $(\infty,1)$-enhancements, see