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arXiv · 2606.00385

Some homological resultsin the category of colour $(A,H)$-Hopf modules

Thomas Guédénon
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Abstract

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 $A^{coH}$.

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Thomas Guédénon. 2026-05-29. Some homological resultsin the category of colour $(A,H)$-Hopf modules. https://arxiv.org/abs/2606.00385

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