Fourier-Mukai functors in the supported case

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.

36 pages. Major revision and new results added. Accepted for publication in Compositio Math

References in corpus (7)

• Uniqueness of enhancement for triangulated categories
• The space of stability conditions on the local projective plane
• Grothendieck ring of pretriangulated categories
• Equivalences of derived categories of sheaves on quasi-projective schemes
• Does full imply faithful?
• The Ringel--Hall Lie algebra of a spherical object
• Sheaves on local Calabi-Yau varieties

Cited by in corpus (7)

• A tour about existence and uniqueness of dg enhancements and lifts
• The uniqueness problem of dg-lifts and Fourier-Mukai kernels
• Scalar extensions of derived categories and non-Fourier-Mukai functors
• Formality and strongly unique enhancements
• Symplectomorphisms and spherical objects in the conifold smoothing
• Non-uniqueness of Fourier-Mukai kernels
• An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves