##  [Fourier-Mukai Transform](/fourier-mukai-transform-1) 

 Definition

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

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

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

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

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

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

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

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

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.