Relative perfect complexes

Abstract

Let f :X rightarrow Y be a morphism of concentrated schemes. We characterize f-perfect complexes {mathscr {E}} as those such that the functor {mathscr {E}}otimes ^{varvec{textsf{L}}}_X varvec{textsf{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.