New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911406298759168 |
|---|---|
| author | Canonaco, Alberto Ornaghi, Mattia |
| author_facet | Canonaco, Alberto Ornaghi, Mattia |
| contents | 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 $\mathbf D^b(\mathbb P^1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra Canonaco, Alberto Ornaghi, Mattia Algebraic Geometry Category Theory Representation Theory 13D09, 14A22, 14F08, 18G35, 18G80 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 $\mathbf D^b(\mathbb P^1)$. |
| title | New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra |
| topic | Algebraic Geometry Category Theory Representation Theory 13D09, 14A22, 14F08, 18G35, 18G80 |
| url | https://arxiv.org/abs/2601.21559 |