Given a Fourier-Mukai functor $Phi$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $Phi$, 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 Anno and Logvinenko, and Hernandez Ruiperez, Lopez Martin and Sancho de Salas, and has applications to the twist autoequivalences of Donovan and Wemyss.
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