Intersection of complete cotorsion pairs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910026197630976 |
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| author | Wang, Qikai Zhu, Haiyan |
| author_facet | Wang, Qikai Zhu, Haiyan |
| contents | Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ is the class of direct summands of extension of $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersection of complete cotorsion pairs Wang, Qikai Zhu, Haiyan K-Theory and Homology 18G25, 18G15, 16E30, 16E10 Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ is the class of direct summands of extension of $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions. |
| title | Intersection of complete cotorsion pairs |
| topic | K-Theory and Homology 18G25, 18G15, 16E30, 16E10 |
| url | https://arxiv.org/abs/2408.01922 |