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Unifying Hopf modules. (English) Zbl 0782.16025
Let $$A$$ be a Hopf algebra over a field $$k$$, $$B$$ a right $$A$$-comodule algebra and $$D$$ a right $$A$$-module coalgebra. In this setup we define the category of $$(D,B)$$-Hopf modules as follows; objects are right $$D$$- comodules with right $$B$$-module structure such that $$\sum(mb)_ 0\otimes(mb)_ 1 = \sum m_ 0b_ 0\otimes m_ 1b_ 1$$ for all $$m$$ in $$M$$ and $$b$$ in $$B$$. (This definition covers the usual Hopf modules.) For any group-like element $$x$$ in $$D$$, we put $$B_ x=\{b\in B; \sum b_ 0\otimes xb_ 1 = b\otimes x\}$$ and $$M_ x = \{m\in M; \sum m_ 0\otimes m_ 1 = m\otimes x\}$$. Then $$M_ x$$ is a right $$B_ x$$-module. We prove that if $$B$$ is a left faithfully flat $$A$$-Galois extension of the co-invariants $$B^{\text{co}A}$$ and the coalgebra map $$A\to D$$, $$a \mapsto xa$$, is left faithfully coflat, then the functor $$M\mapsto M_ x$$ is an equivalence from the category of $$(D,B)$$-Hopf modules to the category of right $$B_ x$$-modules. This result covers several well known equivalences about Hopf modules.
Reviewer: Y.Doi (Fukui)

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16D90 Module categories in associative algebras
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