The <a href="http://en.wikipedia.org/wiki/Mukai-Fourier_transform">Fourier-Mukai transform</a> comes from a bimodule: the Poincaré bundle. Let A be an abelian variety, the Poincaré bundle P is a vector bundle on Ax coming from the fact that the points in the dual abelian variety  parametrize line bundles on A (P is the universal family). In the Fourier-Mukai construction, P is used as a O<sub>A</sub>-O<sub>Â</sub>-bimodule to produce a functor between the derived categories of coherent sheaves on A and  via a push-pull construction.