Paper detail

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, that is, the right adjoint f^\times of the derived functor Rf_* respects small direct sums. This is equivalent to the existence of a functorial isomorphism f^\times O_{Y} \otimes^L Lf^*(-) \to f^\times (-); 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^\times; 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.