Relative perfect complexes

TL;DR

This paper characterizes relative perfect complexes on schemes, showing their stability under certain morphisms, and develops a bivariant theory for their Grothendieck groups, extending classical theorems like semicontinuity and base change.

Contribution

It provides a new characterization of $f$-perfect complexes and extends classical theorems to a relative setting, along with developing a bivariant theory for perfect complexes.

Findings

Characterization of $f$-perfect complexes via tensor and pullback functors.

Quasi-proper morphisms preserve relative perfect complexes.

Generalized semicontinuity and base change theorems for cohomology.

Abstract

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.