Relative perfect complexes

Abstract

Let $$f :X \rightarrow Y$$ f : X → Y be a morphism of concentrated schemes. We characterize f-perfect complexes $${\mathscr {E}}$$ E as those such that the functor $${\mathscr {E}}\otimes ^{\varvec{\textsf{L}}}_X \varvec{\textsf{L}}f^*-$$ E ⊗ X L L f ∗ - preserves bounded complexes. We prove, as a consequence, that a quasi-proper morphism takes relative perfect complexes into perfect ones. We obtain a generalized version of the semicontinuity theorem of dimension of cohomology and Grauert’s base change of the fibers. Finally, a bivariant theory of the Grothendieck group of perfect complexes is developed.