## Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor.(English)Zbl 1124.14003

Summary: 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 $$\mathbf {R}f_\ast$$ respects small direct sums. This is equivalent to the existence of a functorial isomorphism $$f^\times\mathcal O_{Y}\otimes^{\mathbf {L}} \mathbf {L}f^*(-)\overset\sim\longrightarrow f^\times (-)$$; to quasi-properness (preservation by $$\mathbf {R}f$$ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of $$\mathbf {L}f^*$$ (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.

### MSC:

 14A15 Schemes and morphisms 13D02 Syzygies, resolutions, complexes and commutative rings 13D05 Homological dimension and commutative rings 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18E30 Derived categories, triangulated categories (MSC2010)
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