An integral functor between bounded derived categories of coherent sheaves on algebraic varieties defined by an object (the kernel) on the product, which frequently yields equivalences and transports geometric information through derived categories.

Principle

The organizing rule is that a kernel object on X×Y defines a functor D^b(Coh(X)) → D^b(Coh(Y)) by pulling to the product, tensoring with the kernel, and pushing forward; when this functor is an equivalence (or fully faithful), it encodes a deep relation between the geometries of X and Y.

Demonstration

Classical example: the Poincaré line bundle on A× induces the Fourier–Mukai equivalence between the derived categories of a complex abelian variety A and its dual Â; other demonstrations include derived equivalences induced by kernels associated to flops or moduli spaces of stable sheaves.

Misapplication

Applying the naive push–pull recipe outside the derived setting or with kernels that do not satisfy required finiteness or flatness conditions; assuming any exact functor between derived categories is given by a Fourier–Mukai kernel without verifying representability (counterexamples exist in non-projective contexts).

Consequence

When present, a Fourier–Mukai transform translates cohomological and categorical invariants between varieties, gives rise to derived invariants of birational geometry, and underlies many dualities and moduli identifications in algebraic geometry.

Reversal

The inverse (or adjoint) transform is given by a kernel on Y×X related by derived duality to the original kernel; reversing swaps the roles of source and target and corresponds geometrically to the inverse relation between the varieties when an equivalence holds.

Boundary

Most robustly formulated for smooth projective varieties and bounded derived categories of coherent sheaves; requires appropriate properness, finite Tor-dimension and cohomological finiteness for kernels, and does not generically apply to arbitrary triangulated categories or non-coherent sheaf contexts.

Semantic Tension

Tension arises between calling it a 'transform' (suggesting a functorial construction that may be noninvertible) and an 'equivalence' (a two-way categorical isomorphism): some kernels produce equivalences while others give only fully faithful or merely interesting functors, so the terminology masks a spectrum of behaviors.

Synthesis

A Fourier–Mukai Transform is the derived integral transform defined by a kernel on the product of varieties; under finiteness and duality conditions it yields equivalences of derived categories that reflect and relate the underlying algebraic geometries.