Gorenstein cohomological dimension and stable categories for groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916488739291136 |
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| author | Ren, Wei |
| author_facet | Ren, Wei |
| contents | First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in literature obtained over the coefficient ring $\mathbb{Z}$ or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category $\mathcal{F}ib$ consisting of fibrant $RG$-modules, and show that the homotopy category $\mathrm{Ho}(\mathcal{F}ib)$ is triangle equivalent to both the stable category $\underline{\mathcal{C}of}(RG)$ of Benson's cofibrant modules, and the stable module category ${\rm StMod}(RG)$. The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that ${\rm Gcd}_RG<\infty$, ${\rm Ho}(\mathcal{F}ib)$ is equivalent to the stable category of Gorenstein projective $RG$-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective $RG$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09589 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gorenstein cohomological dimension and stable categories for groups Ren, Wei K-Theory and Homology 18G20, 18E30, 20C07, 55U35 First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in literature obtained over the coefficient ring $\mathbb{Z}$ or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category $\mathcal{F}ib$ consisting of fibrant $RG$-modules, and show that the homotopy category $\mathrm{Ho}(\mathcal{F}ib)$ is triangle equivalent to both the stable category $\underline{\mathcal{C}of}(RG)$ of Benson's cofibrant modules, and the stable module category ${\rm StMod}(RG)$. The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that ${\rm Gcd}_RG<\infty$, ${\rm Ho}(\mathcal{F}ib)$ is equivalent to the stable category of Gorenstein projective $RG$-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective $RG$-modules. |
| title | Gorenstein cohomological dimension and stable categories for groups |
| topic | K-Theory and Homology 18G20, 18E30, 20C07, 55U35 |
| url | https://arxiv.org/abs/2206.09589 |