On the existence of Fourier–Mukai functors

A theorem by Orlov states that any equivalence \(F:D^{b}_{\mathrm {Coh}}(X) \rightarrow D^{b}_{\mathrm {Coh}}(Y)\) between the bounded derived categories of coherent sheaves of two smooth projective varieties X and Y is isomorphic to a Fourier–Mukai transform \(\Phi _{E}(-)=R\pi _{2*}(E\mathop {\otimes }\limits ^{L} L\pi _1^{*}(-))\), where the kernel E is in \(D^{b}_{\mathrm {Coh}}(X\times Y)\). In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism \(F\cong \Phi _{E}\) exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result.