2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102079We 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 contemplate a criterion for equivalence. We offer a proof that is new even if we restrict to the smooth case. 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 also is it. These results imply the invertibility of the usual relative Fourier-Mukai transform for an elliptic fibration as a direct corollary.Final version. To appear in Advances in MathematicsAlgebraic GeometryCommutative Algebra18E30 (Primary) ; 14F05, 14J27, 14E30, 13D22, 14M05 (Secondary)text