Fourier–Mukai transforms for Gorenstein schemes

We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic. We contemplate a criterion for equivalence as well. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful. These results imply the invertibility of the usual relative Fourier–Mukai transform for an elliptic fibration as a direct corollary.