An integral transform between bounded derived categories of coherent sheaves D^b(X) → D^b(Y) determined by an object (the kernel) P in D^b(X×Y); it sends F ∈ D^b(X) to Rπ_{Y*}(π_X^*F ⊗^L P) and encodes correspondences between X and Y at the derived level.

Principle

Integral kernels compose by derived convolution on fiber products, so composition of Fourier–Mukai transforms corresponds to convolution of kernels; when the kernel satisfies properness and perfectness conditions the transform is exact and can produce equivalences (Fourier–Mukai partners).

Demonstration

Example: for dual abelian varieties A and  the Poincaré line bundle P on A× defines a Fourier–Mukai equivalence D^b(A) ≅ D^b(Â); skyscraper sheaves map to stable vector bundles or translated line bundles according to the kernel, reflecting classical Fourier analysis phenomena in algebraic geometry.

Misapplication

Treating the transform as a pointwise or naive Fourier transform ignoring derived pullback/pushforward, tensor Tor-conditions, or using nonperfect/nonproper kernels that break boundedness and coherence properties; assuming every birational map induces a Fourier–Mukai equivalence without checking kernel existence and finiteness conditions.

Consequence

Provides a powerful source of derived equivalences, identifies moduli spaces, transfers stability conditions and invariants, and yields isomorphisms on Hochschild (co)homology and on many numerical invariants when an equivalence holds.

Reversal

The inverse of a Fourier–Mukai equivalence is again a Fourier–Mukai transform with the adjoint kernel (derived dual and swap); if no adjoint kernel with the required finiteness exists, the transform may be fully faithful but not essentially surjective, producing embeddings rather than equivalences.

Boundary

Works for derived categories of coherent sheaves under hypotheses of properness, finite Tor-dimension and perfect kernels; fails or must be replaced for unbounded categories, noncoherent coefficients, or kernels lacking finiteness, and requires care on singular spaces.

Semantic Tension

Often compared with the classical Fourier transform on function spaces: both use integral kernels and convolution, but Fourier–Mukai lives in derived algebraic geometry and manipulates complexes and sheaf-theoretic pushforward/pullback rather than pointwise oscillatory integrals.

Synthesis

A Fourier–Mukai transform is an integral, kernel-defined functor between derived categories that realizes geometric correspondences as derived convolutions; under finiteness and perfectness hypotheses it yields deep equivalences linking geometry, moduli, and homological invariants.