Fourier-Mukai functors in the supported case

Understand

We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier--Mukai type if the functor satisfies a condition weaker than being fully faithful.

• We also get generalizations of the results in the literature in the case without support conditions.
• Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.