Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor

For a map f : X → Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint f^x of Rf* respects small direct sums. This is equivalent to the existence of a functorial isomorphism f^xO_Y ⊗^L Lf*( - ) →∼ f^x( - ); to quasi-properness (preservation by Rf* of pseudo-coherence, or just properness in the noetherian case) plus boundedness of Lf* (finite tor-dimensionality), or of the functor f^x; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of the local definition of pseudo-coherence of complexes, as well as a refinement of the known fact that the derived category of complexes with quasi-coherent homology is generated by a single perfect complex.