Localizing and colocalizing subcategories on schemes
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915405206913024 |
|---|---|
| author | Alonso, Leovigildo Jeremías, Ana Loureiro, Eduardo |
| author_facet | Alonso, Leovigildo Jeremías, Ana Loureiro, Eduardo |
| contents | A full triangulated subcategory $\mathsf{L} \subset \mathsf{T}$ of triangulated category $\mathsf{T}$ is \emph{localizing} if it is stable for coproducts. If, further, $\mathsf{T}$ is $\otimes$-triangulated, we say that $\mathsf{L}$ is $\otimes$-ideal if $F \otimes G \in \mathsf{L}$ for all $G \in \mathsf{L}$ and all $F \in \mathsf{T}$. Analogously, a full triangulated subcategory $\mathsf{C} \subset \mathsf{T}$ is \emph{colocalizing} if it is stable for products. If, further, $\mathsf{T}$ is \emph{closed}, \textit{i.e.} $\otimes$-triangulated with internal homs (denoted $[-,-]$), we say that $\mathsf{C}$ is $\mathcal{H}$-coideal if $[F, G] \in \mathsf{C}$ for all $G \in \mathsf{C}$ and all $F \in \mathsf{T}$.
For a point generated concentrated scheme $X$, we prove that all $\otimes$-ideal localizing subcategories of $\mathbf{D}_{qc}(X)$ are classified by the subsets of $X$. As a consequence, we prove that for $\mathcal{H}$-coideal colocalizing subcategories of $\mathbf{D}_{qc}(X)$ the same holds. Moreover, every such colocalizing subcategory $\mathsf{C}$ is of the form $\mathsf{C} = \mathsf{L}^\perp$, where $\mathsf{L}$ is a $\otimes$-ideal localizing subcategory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Localizing and colocalizing subcategories on schemes Alonso, Leovigildo Jeremías, Ana Loureiro, Eduardo Algebraic Geometry Category Theory 14F08 (primary), 18E30 (secondary) A full triangulated subcategory $\mathsf{L} \subset \mathsf{T}$ of triangulated category $\mathsf{T}$ is \emph{localizing} if it is stable for coproducts. If, further, $\mathsf{T}$ is $\otimes$-triangulated, we say that $\mathsf{L}$ is $\otimes$-ideal if $F \otimes G \in \mathsf{L}$ for all $G \in \mathsf{L}$ and all $F \in \mathsf{T}$. Analogously, a full triangulated subcategory $\mathsf{C} \subset \mathsf{T}$ is \emph{colocalizing} if it is stable for products. If, further, $\mathsf{T}$ is \emph{closed}, \textit{i.e.} $\otimes$-triangulated with internal homs (denoted $[-,-]$), we say that $\mathsf{C}$ is $\mathcal{H}$-coideal if $[F, G] \in \mathsf{C}$ for all $G \in \mathsf{C}$ and all $F \in \mathsf{T}$. For a point generated concentrated scheme $X$, we prove that all $\otimes$-ideal localizing subcategories of $\mathbf{D}_{qc}(X)$ are classified by the subsets of $X$. As a consequence, we prove that for $\mathcal{H}$-coideal colocalizing subcategories of $\mathbf{D}_{qc}(X)$ the same holds. Moreover, every such colocalizing subcategory $\mathsf{C}$ is of the form $\mathsf{C} = \mathsf{L}^\perp$, where $\mathsf{L}$ is a $\otimes$-ideal localizing subcategory. |
| title | Localizing and colocalizing subcategories on schemes |
| topic | Algebraic Geometry Category Theory 14F08 (primary), 18E30 (secondary) |
| url | https://arxiv.org/abs/2405.10383 |