Categorical Krull-Remak-Schmidt for triangulated categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909183969853440 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in Γ$ and $j \in Λ$. Suppose there is an equivalence of triangulated $R$-categories \[ Φ\colon \bigoplus_{i \in Γ}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in Λ}\mathcal{D}_j \] Then we show that there is a bijective function $π\colon Γ\rightarrow Λ$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{π(i)} $ for all $i \in Γ$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorical Krull-Remak-Schmidt for triangulated categories Puthenpurakal, Tony J. Commutative Algebra Algebraic Geometry Representation Theory Primary 18G80, Secondary 13D09, 13E35 Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in Γ$ and $j \in Λ$. Suppose there is an equivalence of triangulated $R$-categories \[ Φ\colon \bigoplus_{i \in Γ}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in Λ}\mathcal{D}_j \] Then we show that there is a bijective function $π\colon Γ\rightarrow Λ$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{π(i)} $ for all $i \in Γ$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components. |
| title | Categorical Krull-Remak-Schmidt for triangulated categories |
| topic | Commutative Algebra Algebraic Geometry Representation Theory Primary 18G80, Secondary 13D09, 13E35 |
| url | https://arxiv.org/abs/2404.18483 |