Relative PGF modules and dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914912084688896 |
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| author | Maaouy, Rachid El |
| author_facet | Maaouy, Rachid El |
| contents | Inspired in part by recent work of Šaroch and Šťov\'ıček in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod ${\rm G_C}$-dimension of finitely generated modules with respect to a semidualizing module $C$ to arbitrary modules over arbitrary rings, with respect to a module $C$ that is not necessarily semidualizing. We call this dimension ${\rm PG_CF}$ dimension and show that it can serve as an alternative definition of the ${\rm G_C}$-projective dimension introduced by Holm and Jørgensen. Modules with ${\rm PG_CF}$ dimension zero are called ${\rm PG_CF}$ modules. When the module $C$ is nice enough, we show that the class ${\rm PG_CF}(R)$ of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that ${\rm PG_CF}(R)$ is the left-hand side of a complete hereditary cotorsion pair. This yields, from a homotopical perspective, a hereditary Hovey triple where the cofibrant objects coincide with the ${\rm PG_CF}$ modules and the fibrant objects coincide with the modules in the well-known Bass class $\mathcal{B}_C(R)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_07232 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative PGF modules and dimensions Maaouy, Rachid El Rings and Algebras Representation Theory 18N40, 18G20, 18G25 Inspired in part by recent work of Šaroch and Šťov\'ıček in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod ${\rm G_C}$-dimension of finitely generated modules with respect to a semidualizing module $C$ to arbitrary modules over arbitrary rings, with respect to a module $C$ that is not necessarily semidualizing. We call this dimension ${\rm PG_CF}$ dimension and show that it can serve as an alternative definition of the ${\rm G_C}$-projective dimension introduced by Holm and Jørgensen. Modules with ${\rm PG_CF}$ dimension zero are called ${\rm PG_CF}$ modules. When the module $C$ is nice enough, we show that the class ${\rm PG_CF}(R)$ of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that ${\rm PG_CF}(R)$ is the left-hand side of a complete hereditary cotorsion pair. This yields, from a homotopical perspective, a hereditary Hovey triple where the cofibrant objects coincide with the ${\rm PG_CF}$ modules and the fibrant objects coincide with the modules in the well-known Bass class $\mathcal{B}_C(R)$. |
| title | Relative PGF modules and dimensions |
| topic | Rings and Algebras Representation Theory 18N40, 18G20, 18G25 |
| url | https://arxiv.org/abs/2408.07232 |