Let $f \colon X \to Y$ be a morphism of concentrated schemes. We characterize $f$-perfect complexes $\mathcal{ E}$ as those such that the functor $\mathcal{ E} \otimes^{ \mathbf{ L}}_X \mathbf{ L} f^*-$ 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.
Categories: math.AG