Quasi-resolving subcategories and dimensions in extriangulated categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914194921619456 |
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| author | He, Zhenggang Shi, Longfu Li, Shuangyan |
| author_facet | He, Zhenggang Shi, Longfu Li, Shuangyan |
| contents | Let $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we introduce and study quasi-resolving subcategories in $\mathcal{C}$. More precisely, we first introduce the notion of $\mathcal{X}$-resolution dimensions for a quasi-resolving subcategory $\mathcal{X}$ of $\mathcal{C}$ and then give some equivalent characterizations of objects which have finite $\mathcal{X}$-resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by $\mathcal{GQP}_{\mathcal{X}}(ξ)$, in term of a quasi-resolving subcategory $\mathcal{X}$, and prove that $\mathcal{GQP}_{\mathcal{X}}(ξ)$ is also a quasi-resolving subcategory of $\mathcal{C}$. Moreover, some classical known results are generalized in $\mathcal{GQP}_{\mathcal{X}}(ξ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-resolving subcategories and dimensions in extriangulated categories He, Zhenggang Shi, Longfu Li, Shuangyan Category Theory Let $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we introduce and study quasi-resolving subcategories in $\mathcal{C}$. More precisely, we first introduce the notion of $\mathcal{X}$-resolution dimensions for a quasi-resolving subcategory $\mathcal{X}$ of $\mathcal{C}$ and then give some equivalent characterizations of objects which have finite $\mathcal{X}$-resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by $\mathcal{GQP}_{\mathcal{X}}(ξ)$, in term of a quasi-resolving subcategory $\mathcal{X}$, and prove that $\mathcal{GQP}_{\mathcal{X}}(ξ)$ is also a quasi-resolving subcategory of $\mathcal{C}$. Moreover, some classical known results are generalized in $\mathcal{GQP}_{\mathcal{X}}(ξ)$. |
| title | Quasi-resolving subcategories and dimensions in extriangulated categories |
| topic | Category Theory |
| url | https://arxiv.org/abs/2511.07039 |