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

Abstract

For a map $f\colon X\to Y$ of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint $f^\times$ of $\mathbf Rf_*$ respects small direct sums. This is equivalent to the existence of a functorial isomorphism $f^\times\mathcal O_{Y}\otimes^{\mathbf L} \mathbf Lf^*(\<-\<)\! {\longrightarrow{}^\sim} f^\times (-)$; to quasi-properness (preservation by $\Rf$ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of $\mathbf Lf^*\<$ (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.