Flat model structures and Gorenstein objects in functor categories
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912969681534976 |
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| author | Di, Zhenxing Li, Liping Liang, Li Ma, Yajun |
| author_facet | Di, Zhenxing Li, Liping Liang, Li Ma, Yajun |
| contents | We construct a flat model structure on the category $_{\mathcal{Q},R}{\mathsf{Mod}}$ of additive functors from a small preadditive category $\mathcal{Q}$ satisfying certain conditions to the module category $_{R}{\mathsf{Mod}}$ over an associative ring $R$, whose homotopy category is the $\mathcal{Q}$-shaped derived category introduced by Holm and Jorgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in $_{\mathcal{Q},R}{\mathsf{Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal{Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and Šťov\'ıček. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10945 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Flat model structures and Gorenstein objects in functor categories Di, Zhenxing Li, Liping Liang, Li Ma, Yajun Representation Theory Rings and Algebras We construct a flat model structure on the category $_{\mathcal{Q},R}{\mathsf{Mod}}$ of additive functors from a small preadditive category $\mathcal{Q}$ satisfying certain conditions to the module category $_{R}{\mathsf{Mod}}$ over an associative ring $R$, whose homotopy category is the $\mathcal{Q}$-shaped derived category introduced by Holm and Jorgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in $_{\mathcal{Q},R}{\mathsf{Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal{Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and Šťov\'ıček. |
| title | Flat model structures and Gorenstein objects in functor categories |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2211.10945 |