Ideal approximation theory in Frobenius categories
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866915154248073216 |
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| author | Sun, Dandan Tan, Zhongsheng Wang, Qikai Zhu, Haiyan |
| author_facet | Sun, Dandan Tan, Zhongsheng Wang, Qikai Zhu, Haiyan |
| contents | Let $\mathcal{A}$ be a Frobenius category and $ω$ the full subcategory consisting of projective objects. The relations between special precovering (resp., precovering) ideals in $\mathcal{A}$ and special precovering (resp., preenveloping) ideals in the stable category $\mathcal{A}/ω$ are explored. In combination with a result due to Breaz and Modoi, we conclude that every precovering or preenveloping ideal $\mathcal{I}$ in $\mathcal{A}$ with $1_{X}\in{\mathcal{I}}$ for any $X\inω$ is special. As a consequence, it is proved that an ideal cotorsion pair $(\mathcal{I},\mathcal{J})$ in $\mathcal{A}$ is complete if and only if $\mathcal{I}$ is precovering if and only if $\mathcal{J}$ is preenveloping. This leads to an ideal version of the Bongartz-Eklof-Trlifaj Lemma in $\mathcal{A}/ω$, which states that an ideal cotorsion pair in $\mathcal{A}/ω$ generated by a set of morphisms is complete. As another consequence, we provide some partial answers to the question about the completeness of cotorsion pairs posed by Fu, Guil Asensio, Herzog and Torrecillas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ideal approximation theory in Frobenius categories Sun, Dandan Tan, Zhongsheng Wang, Qikai Zhu, Haiyan Category Theory 18G10, 18G25, 16D90 Let $\mathcal{A}$ be a Frobenius category and $ω$ the full subcategory consisting of projective objects. The relations between special precovering (resp., precovering) ideals in $\mathcal{A}$ and special precovering (resp., preenveloping) ideals in the stable category $\mathcal{A}/ω$ are explored. In combination with a result due to Breaz and Modoi, we conclude that every precovering or preenveloping ideal $\mathcal{I}$ in $\mathcal{A}$ with $1_{X}\in{\mathcal{I}}$ for any $X\inω$ is special. As a consequence, it is proved that an ideal cotorsion pair $(\mathcal{I},\mathcal{J})$ in $\mathcal{A}$ is complete if and only if $\mathcal{I}$ is precovering if and only if $\mathcal{J}$ is preenveloping. This leads to an ideal version of the Bongartz-Eklof-Trlifaj Lemma in $\mathcal{A}/ω$, which states that an ideal cotorsion pair in $\mathcal{A}/ω$ generated by a set of morphisms is complete. As another consequence, we provide some partial answers to the question about the completeness of cotorsion pairs posed by Fu, Guil Asensio, Herzog and Torrecillas. |
| title | Ideal approximation theory in Frobenius categories |
| topic | Category Theory 18G10, 18G25, 16D90 |
| url | https://arxiv.org/abs/2502.11146 |