Non-uniqueness of Fourier-Mukai kernels

TL;DR

This paper demonstrates that Fourier-Mukai kernels are generally not unique, but their cohomology sheaves are, and explores properties of the associated functor between derived categories of smooth projective schemes.

Contribution

It establishes the non-uniqueness of Fourier-Mukai kernels while confirming the uniqueness of their cohomology sheaves, and analyzes related functor properties.

Findings

Fourier-Mukai kernels are not unique in general.

Cohomology sheaves of kernels are unique.

Properties of the functor from derived categories to Fourier-Mukai functors are discussed.

Abstract

We prove that the kernels of Fourier-Mukai functors are not unique in general. On the other hand we show that the cohomology sheaves of those kernels are unique. We also discuss several properties of the functor sending an object in the derived category of the product of two smooth projective schemes to the corresponding Fourier-Mukai functor.