On the class of Benson's cofibrant modules
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912262327894016 |
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| author | Emmanouil, Ioannis Ren, Wei |
| author_facet | Emmanouil, Ioannis Ren, Wei |
| contents | In this paper, we examine the class of cofibrant modules over a group algebra $kG$, that were defined by Benson in [2]. We show that this class is always the left-hand side of a complete hereditary cotorsion pair in the category of $kG$-modules. It follows that the class of Gorenstein projective $kG$-modules is special precovering in the category of $kG$-modules, if $G$ is contained in the class ${\scriptstyle{\bf LH}}\mathfrak{F}$ of hierarchically decomposable groups defined by Kropholler in [20] and $k$ has finite weak global dimension. It also follows that the obstruction to the equality between the classes of cofibrant and Gorenstein projective $kG$-modules can be described, over any group algebra $kG$, in terms of a suitable subcategory of the stable category of Gorenstein projective $kG$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the class of Benson's cofibrant modules Emmanouil, Ioannis Ren, Wei K-Theory and Homology In this paper, we examine the class of cofibrant modules over a group algebra $kG$, that were defined by Benson in [2]. We show that this class is always the left-hand side of a complete hereditary cotorsion pair in the category of $kG$-modules. It follows that the class of Gorenstein projective $kG$-modules is special precovering in the category of $kG$-modules, if $G$ is contained in the class ${\scriptstyle{\bf LH}}\mathfrak{F}$ of hierarchically decomposable groups defined by Kropholler in [20] and $k$ has finite weak global dimension. It also follows that the obstruction to the equality between the classes of cofibrant and Gorenstein projective $kG$-modules can be described, over any group algebra $kG$, in terms of a suitable subcategory of the stable category of Gorenstein projective $kG$-modules. |
| title | On the class of Benson's cofibrant modules |
| topic | K-Theory and Homology |
| url | https://arxiv.org/abs/2503.04284 |