Fourier-Mukai functors in the supported case

TL;DR

This paper demonstrates that certain exact functors between categories of supported perfect complexes on projective schemes are of Fourier--Mukai type under weaker conditions than full faithfulness, with applications and uniqueness results.

Contribution

It extends Fourier--Mukai theory to supported complexes with weaker assumptions and establishes the strong uniqueness of their enhancements.

Findings

Exact functors are of Fourier--Mukai type under weaker conditions.

Generalizations to non-support cases are provided.

The category of supported perfect complexes has a strongly unique enhancement.

Abstract

We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier--Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the case without support conditions. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.