Relative integral functors for singular fibrations and singular partners

We study relative integral functors for singular schemes and characterise those which preserve boundness and those which have integral right adjoints. We prove that a relative integral functor is an equivalence if and only if its restriction to every bre is an equivalence. This allows us to construct a non-trivial auto-equivalence of the derived category of an arbitrary genus one bration with no conditions on either the base or the total space and getting rid of the usual assumption of irreducibility of the bres. We also extend to Cohen-Macaulay schemes the criterion of Bondal and Orlov for an integral functor to be fully faithful in characteristic zero and give a dierent criterion which is valid in arbitrary characteristic. Finally, we prove that for projective schemes both the Cohen-Macaulay and the Gorenstein conditions are invariant under Fourier-Mukai functors.