Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Injectives obstruct Fourier-Mukai functors

Felix Küng

TL;DR

The work develops a novel obstruction framework for lifting exact functors between derived categories to the dg/\text{A}_\infty level by replacing tilting objects with the subcategory of injectives. By computing the Inj-equivariant characteristic morphism and tracking obstructions through a divisor-based pushforward, the authors construct explicit non-Fourier-Mukai functors $\psi_\eta$, and show that for smooth hypersurfaces there exists a large space of such functors. The results yield non-liftability without requiring tilting objects and provide concrete, computable obstructions, including explicit twisted-Hodge-diamond data for divisors. The methodology potentially extends non-liftability phenomena to coherent sheaves and clarifies the landscape of functors beyond the Fourier-Mukai paradigm in geometric settings. Overall, the paper broadens the toolkit for distinguishing FM and non-FM functors via Hochschild-obstruction theory in a dg/$\text{A}_\infty$ context.

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.

Injectives obstruct Fourier-Mukai functors

TL;DR

The work develops a novel obstruction framework for lifting exact functors between derived categories to the dg/\text{A}_\infty level by replacing tilting objects with the subcategory of injectives. By computing the Inj-equivariant characteristic morphism and tracking obstructions through a divisor-based pushforward, the authors construct explicit non-Fourier-Mukai functors , and show that for smooth hypersurfaces there exists a large space of such functors. The results yield non-liftability without requiring tilting objects and provide concrete, computable obstructions, including explicit twisted-Hodge-diamond data for divisors. The methodology potentially extends non-liftability phenomena to coherent sheaves and clarifies the landscape of functors beyond the Fourier-Mukai paradigm in geometric settings. Overall, the paper broadens the toolkit for distinguishing FM and non-FM functors via Hochschild-obstruction theory in a dg/ context.

Abstract

We use injectives as a big tilting object to obstruct liftability of exact functors to the -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 /-lift. This approach allows explicit computation of the obstruction against an -lift. We in particular observe that this computation gives for smooth degree hypersurfaces an abundance of non-Fourier-Mukai functors.
Paper Structure (13 sections, 43 theorems, 132 equations)

This paper contains 13 sections, 43 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Proof strategy
  3. Structure of the paper
  4. Preliminaries
  5. Hochschild cohomology and characteristic morphism
  6. Equivariant objects over categories
  7. Equivariant characteristic morphism of Injectives
  8. Interpreting characteristic morphisms of a subcategory as pullback
  9. The $\mathrm{Inj}$-equivariant characteristic morphism
  10. Construction of $\psi_\eta$
  11. Recollection on Obstructions
  12. Structure of the functor $\widetilde{f}$
  13. Computing obstructions explicitly for Divisors

Key Result

Theorem 1.2

Orlov1997 Let $f:\mathcal{D}^b\left(\mathrm{coh} X\right) \to \mathcal{D}^b\left(\mathrm{coh} Y\right)$ be a fullyfaithful functor between derived categories of smooth projective schemes. Then $f$ is Fourier-Mukai.

Theorems & Definitions (90)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 8.11
  • Corollary 8.13
  • Theorem 8.12
  • Definition 2.1
  • Lemma 2.2
  • ...and 80 more