This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds of cohomology theories for many decades. In fact, Orlov's celebrated result, which is stated but not proved, says that any equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type. Fourier-Mukai functors behave well in many respects: they are exact, admit left and right adjoints, can be composed, etc. The cohomological Fourier-Mukai transform behaves with respect to grading, Hodge structure, and Mukai pairing.
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