In algebraic geometry, a Fourier–Mukai transform ΦK  is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y).
In algebraic geometry, the Fourier–Mukai transform ΦK is a mapping between the categories of coherent sheaves on two schemes X and Y, which can be understood as an integral transformation that relates the two schemes through a kernel object K in the derived category of their product scheme.