Gorenstein homological invariants and monoidal model categories of Hopf algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918310662111232 |
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| author | Ren, Wei Zhu, Ruipeng |
| author_facet | Ren, Wei Zhu, Ruipeng |
| contents | Let $H$ be a Hopf algebra over a field $k$ with a bijective antipode. It is proved that the Gorenstein global dimension of $H$ coincides with the Gorenstein projective dimension of the trivial left (or right) $H$-module $k$. Then, $H$ is finite dimensional if and only if the Gorenstein projective dimension of $k$ is trivial. Although monoidal Morita-Takeuchi equivalence of Hopf algebras does not preserve the global dimension, we demonstrate that it does preserve the Gorenstein global dimension and the Artin-Schelter Gorenstein property; this supports Brown-Goodearl's question of whether every noetherian (affine) Hopf algebra is AS Gorenstein. Finally, for $H$ and an $H$-Galois object $B$, we show the categories of modules $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$ are monoidal model categories regarding Gorenstein projective model structure, provided that the Gorenstein global dimension of $H$ is finite. The corresponding stable categories are tensor triangulated categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20249 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gorenstein homological invariants and monoidal model categories of Hopf algebras Ren, Wei Zhu, Ruipeng Rings and Algebras K-Theory and Homology Quantum Algebra 16E10, 16T05, 16E65, 18N40 Let $H$ be a Hopf algebra over a field $k$ with a bijective antipode. It is proved that the Gorenstein global dimension of $H$ coincides with the Gorenstein projective dimension of the trivial left (or right) $H$-module $k$. Then, $H$ is finite dimensional if and only if the Gorenstein projective dimension of $k$ is trivial. Although monoidal Morita-Takeuchi equivalence of Hopf algebras does not preserve the global dimension, we demonstrate that it does preserve the Gorenstein global dimension and the Artin-Schelter Gorenstein property; this supports Brown-Goodearl's question of whether every noetherian (affine) Hopf algebra is AS Gorenstein. Finally, for $H$ and an $H$-Galois object $B$, we show the categories of modules $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$ are monoidal model categories regarding Gorenstein projective model structure, provided that the Gorenstein global dimension of $H$ is finite. The corresponding stable categories are tensor triangulated categories. |
| title | Gorenstein homological invariants and monoidal model categories of Hopf algebras |
| topic | Rings and Algebras K-Theory and Homology Quantum Algebra 16E10, 16T05, 16E65, 18N40 |
| url | https://arxiv.org/abs/2601.20249 |