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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2606.00385 |
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| _version_ | 1866914619614822400 |
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| author | Guédénon, Thomas |
| author_facet | Guédénon, Thomas |
| contents | Let $\Bbbk$ be a field, $H$ a colour Hopf algebra and $A$ a graded $H$-comodule colour algebra. We give a sufficient condition for a colour $(A,H)$-Hopf module to be injective as a graded $H$-comodule and we deduce relative projectivity in the category of colour $(A,H)$-Hopf modules. We generalize the Fundamental Theorem of $(A,H)$-Hopf modules to the context of colour $(A,H)$-Hopf modules. Using this result, we show that the categories of graded $A^{coH}$-modules and of colour $(A,H)$-Hopf modules are equivalent, $A$ is faithfully flat as a graded right $A^{coH}$-module and is a graded Hopf-Galois extension of $A^{coH}$. Under some assumptions, we show that $M^{coH}$ is a graded $A$-module and we prove that the graded global dimension of $A$ is equal to the graded projective dimension of the graded $A$-module $A^{coH}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00385 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some homological resultsin the category of colour $(A,H)$-Hopf modules Guédénon, Thomas Rings and Algebras 16T05, 16D40 Let $\Bbbk$ be a field, $H$ a colour Hopf algebra and $A$ a graded $H$-comodule colour algebra. We give a sufficient condition for a colour $(A,H)$-Hopf module to be injective as a graded $H$-comodule and we deduce relative projectivity in the category of colour $(A,H)$-Hopf modules. We generalize the Fundamental Theorem of $(A,H)$-Hopf modules to the context of colour $(A,H)$-Hopf modules. Using this result, we show that the categories of graded $A^{coH}$-modules and of colour $(A,H)$-Hopf modules are equivalent, $A$ is faithfully flat as a graded right $A^{coH}$-module and is a graded Hopf-Galois extension of $A^{coH}$. Under some assumptions, we show that $M^{coH}$ is a graded $A$-module and we prove that the graded global dimension of $A$ is equal to the graded projective dimension of the graded $A$-module $A^{coH}$. |
| title | Some homological resultsin the category of colour $(A,H)$-Hopf modules |
| topic | Rings and Algebras 16T05, 16D40 |
| url | https://arxiv.org/abs/2606.00385 |