The uniqueness problem of dg-lifts and Fourier-Mukai kernels (Q2835339)

The paper under review discusses the relationship between functors between triangulated categories, and quasifunctors between their enhancements. The \textit{uniqueness problem} asks for conditions under which two two quasi-functors between pretriangulated dg categories are isomorphic, provided that their induced functors on the homotopy categories are. Inspired by results for Fourier-Mukai transforms a general criterion in the categorical setting is shown, which moreover implies (and strengthens) the uniqueness results for Fourier-Mukai transforms.NEWLINENEWLINEIn the geometric setting, the first uniqueness result for Fourier-Mukai kernels is for fully faithful functors between derived categories of smooth projective varieties, due to Bondal-Orlov. This was improved by \textit{A.\ Canonaco} and \textit{P.\ Stellari} [Adv. Math. 212, No. 2, 484--503 (2007; Zbl 1116.14009)], introducing a condition involving the vanishing of negative Ext's between the images of coherent sheaves under the Fourier-Mukai kernel.NEWLINENEWLINEIn the categorical setting, the paper shows that if the domain of the quasi-functors is the pretriangulated hull of a \(k\)-linear category, and there are no negative Ext's between the images of objects in this linear category, then we can lift an isomorphism of exact functors to an isomorphism of quasi-functors. The proof uses the obstruction theory for morphisms in \(\text{A}_\infty\)-categories, and inductively constructs the isomorphism of quasi-functors. Back in the geometric setting it is possible to apply this to every quasiprojective scheme, by using an ample line bundle to construct a \(k\)-linear dg enhancement.