Adjoints to a Fourier-Mukai functor

Given a Fourier-Mukai functor Φ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to Φ, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier-Mukai kernel is a perfect complex. This extends previous work of [1,12,13] and has applications to the twist autoequivalences of [9].