New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra
Alberto Canonaco, Mattia Ornaghi
TL;DR
This work addresses the existence of non-Fourier-Mukai exact functors between derived categories in low dimensions. It develops an algebraic framework that identifies exact functors with good octahedra in the target triangulated category, and shows obstruction phenomena when two nonisomorphic good octahedra extend the same pair of morphisms. By constructing a simple three-object exceptional sequence and two nonisomorphic good octahedra, the authors produce a nonliftable exact functor to $\\mathbf D^b(\\Bbbk[x]-\\mathrm{mod})$, which they then transport to geometry to obtain explicit non-Fourier-Mukai functors between $\\mathbf D^b(\\mathbb P^2)$ and $\\mathbf D^b(\\mathbb P^1)$. The results demonstrate that even in dimension two and over arbitrary fields, there exist explicit non-FM functors and clarify the role of dg lifts and enhancements in Fourier-Mukai representability.
Abstract
We study a triangulated category $\mathscr S$ that admits a full and strong exceptional sequence of three objects with one-dimensional Hom spaces. We show that the isomorphism classes of exact functors from $\mathscr S$ to another triangulated category $\mathscr T$ are in bijection with the isomorphism classes of octahedra in $\mathscr T$ satisfying a natural condition. As an application, we construct an exact functor from $\mathscr S$ to $\mathbf D^b(\Bbbk[x]\text{-}\mathrm{mod})$ that does not admit a dg lift. This provides an explicit example of a non-Fourier-Mukai exact functor between $\mathbf D^b(\mathbb P^2)$ and $\mathbf D^b(\mathbb P^1)$.