Injectives obstruct Fourier-Mukai functors

TL;DR

This paper introduces a method using injectives as tilting objects to identify and compute obstructions to lifting exact functors to the dg-level, revealing many non-Fourier-Mukai functors for certain hypersurfaces.

Contribution

It presents a novel approach employing injectives to obstruct liftability of functors, providing explicit obstruction computations for non-Fourier-Mukai functors.

Findings

Many non-Fourier-Mukai functors for smooth degree d>2 hypersurfaces

Explicit obstruction computations for dg-lifts

Injectives as tilting objects facilitate non-liftability proofs

Abstract

We use injectives as a big tilting object to obstruct liftability of exact functors to the $\dg$-level. We use the inclusion of injectives into the canonical heart as a replacement for tilting objects in computations of the characteristic morphism. Then we apply this construction to proofs of non-liftability of candidate non-Fourier-Mukai functors, i.e.\ functors that do not admit an $\Ainfty$/$\dg$-lift. This approach allows explicit computation of the obstruction against an $\Ainfty$-lift. We in particular observe that this computation gives for smooth degree $d>2$ hypersurfaces an abundance of non-Fourier-Mukai functors.