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 of Rf∗ respects small direct sums. This is equivalent to the existence of a functorial isomorphism f×OY ⊗ L Lf(− ) −→ ∼ f(−); 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; 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.