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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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