Cotorsion pairs in extensions of abelian categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915357235609600 |
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| author | Hu, Dongdong |
| author_facet | Hu, Dongdong |
| contents | Let $\mathcal{B}$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal{A}=\mathcal{B}\ltimes_η\mathsf{F}$ be an $η$-extension of $\mathcal{B}$. Given a cotorsion pair $(\mathcal{X},\;\mathcal{Y})$ in $\mathcal{B}$, we construct a cotorsion pair $(^{\perp}{\mathsf{U}^{-1}(\mathcal{Y})}, \mathsf{U}^{-1}(\mathcal{Y}))$ in $\mathcal{A}$ and a cotorsion pair $(Δ(\mathcal{X}),\;Δ(\mathcal{X})^\perp)$ in $\mathcal{A}$ for $\mathsf{F}^2=0$. In addition, the heredity and completeness of these cotorsion pairs are studied. Finally, we give some applications and examples in comma categories, some Morita context rings and trivial extensions of rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cotorsion pairs in extensions of abelian categories Hu, Dongdong Representation Theory 16B50, 16E30, 18A25, 18G25 Let $\mathcal{B}$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal{A}=\mathcal{B}\ltimes_η\mathsf{F}$ be an $η$-extension of $\mathcal{B}$. Given a cotorsion pair $(\mathcal{X},\;\mathcal{Y})$ in $\mathcal{B}$, we construct a cotorsion pair $(^{\perp}{\mathsf{U}^{-1}(\mathcal{Y})}, \mathsf{U}^{-1}(\mathcal{Y}))$ in $\mathcal{A}$ and a cotorsion pair $(Δ(\mathcal{X}),\;Δ(\mathcal{X})^\perp)$ in $\mathcal{A}$ for $\mathsf{F}^2=0$. In addition, the heredity and completeness of these cotorsion pairs are studied. Finally, we give some applications and examples in comma categories, some Morita context rings and trivial extensions of rings. |
| title | Cotorsion pairs in extensions of abelian categories |
| topic | Representation Theory 16B50, 16E30, 18A25, 18G25 |
| url | https://arxiv.org/abs/2506.17898 |