Support $τ$-tilting subcategories in exact categories
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910309034229760 |
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| author | Pan, Jixing Zhang, Yaohua Zhu, Bin |
| author_facet | Pan, Jixing Zhang, Yaohua Zhu, Bin |
| contents | Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $τ$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support $τ$-tilting modules (subcategories) in various context. It is also a generalization of tilting subcategories of exact categories. We show that there is a bijection between support $τ$-tilting subcategories and certain $τ$-cotorsion pairs. Given a support $τ$-tilting subcategory $\mathcal{T}$, we find a subcategory $\mathcal{E}_{\mathcal{T}}$ of $\mathcal{E}$ which is an exact category and $\mathcal{T}$ is a tilting subcategory of $\mathcal{E}_{\mathcal{T}}$. If $\mathcal{E}$ is Krull-Schmidt, we prove the cardinal $|\mathcal{T}|$ is equal to the number of isomorphism classes of indecomposable projectives $Q$ such that ${\rm Hom}_{\mathcal{E}}(Q,\mathcal{T})\neq 0$. We also show a functorial version of Brenner-Butler's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10437 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Support $τ$-tilting subcategories in exact categories Pan, Jixing Zhang, Yaohua Zhu, Bin Representation Theory 16G10, 18E40, 16S90, 18E99 Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $τ$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support $τ$-tilting modules (subcategories) in various context. It is also a generalization of tilting subcategories of exact categories. We show that there is a bijection between support $τ$-tilting subcategories and certain $τ$-cotorsion pairs. Given a support $τ$-tilting subcategory $\mathcal{T}$, we find a subcategory $\mathcal{E}_{\mathcal{T}}$ of $\mathcal{E}$ which is an exact category and $\mathcal{T}$ is a tilting subcategory of $\mathcal{E}_{\mathcal{T}}$. If $\mathcal{E}$ is Krull-Schmidt, we prove the cardinal $|\mathcal{T}|$ is equal to the number of isomorphism classes of indecomposable projectives $Q$ such that ${\rm Hom}_{\mathcal{E}}(Q,\mathcal{T})\neq 0$. We also show a functorial version of Brenner-Butler's theorem. |
| title | Support $τ$-tilting subcategories in exact categories |
| topic | Representation Theory 16G10, 18E40, 16S90, 18E99 |
| url | https://arxiv.org/abs/2301.10437 |