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 {E}}\otimes ^{\varvec{\textsf{L}}}_X \varvec{\textsf{L}}f^*-$$ ⊗ L ∗ - preserves bounded complexes. prove, consequence, quasi-proper takes relative perfect into ones. obtain generalized version semicontinuity theorem dimension cohomology and Grauert’s base change fibers. Finally, bivariant theory Grothendieck group is developed.