Classification of localizing subcategories along t-structures
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915061306490880 |
|---|---|
| author | Aambø, Torgeir |
| author_facet | Aambø, Torgeir |
| contents | We study the interplay between localizing subcategories in a stable $\infty$-category $\mathcal{C}$ with $t$-structure $(\mathcal{C}_{\geq 0}, \mathcal{C}_{\leq 0})$, the prestable $\infty$-category $\mathcal{C}_{\geq 0}$ and the abelian category $\mathcal{C}^{\heartsuit}$. We prove that weak localizing subcategories of $\mathcal{C}^{\heartsuit}$ are in bijection with the localizing subcategories of $\mathcal{C}$ where object-containment can be checked on the heart. This generalizes similar known correspondences for noetherian rings and bounded $t$-structures. We also prove that this restricts to a bijection between localizing subcategories of $\mathcal{C}^{\heartsuit}$, and localizing subcategories of $\mathcal{C}$ that are kernels of $t$-exact functors -- lifting Lurie's correspondence between localizing subcategories in $\mathcal{C}_{\geq 0}$ and $\mathcal{C}^{\heartsuit}$ to the stable category $\mathcal{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of localizing subcategories along t-structures Aambø, Torgeir Algebraic Topology Category Theory We study the interplay between localizing subcategories in a stable $\infty$-category $\mathcal{C}$ with $t$-structure $(\mathcal{C}_{\geq 0}, \mathcal{C}_{\leq 0})$, the prestable $\infty$-category $\mathcal{C}_{\geq 0}$ and the abelian category $\mathcal{C}^{\heartsuit}$. We prove that weak localizing subcategories of $\mathcal{C}^{\heartsuit}$ are in bijection with the localizing subcategories of $\mathcal{C}$ where object-containment can be checked on the heart. This generalizes similar known correspondences for noetherian rings and bounded $t$-structures. We also prove that this restricts to a bijection between localizing subcategories of $\mathcal{C}^{\heartsuit}$, and localizing subcategories of $\mathcal{C}$ that are kernels of $t$-exact functors -- lifting Lurie's correspondence between localizing subcategories in $\mathcal{C}_{\geq 0}$ and $\mathcal{C}^{\heartsuit}$ to the stable category $\mathcal{C}$. |
| title | Classification of localizing subcategories along t-structures |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2412.09391 |