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On modules induced or coinduced from Hopf subalgebras. (English) Zbl 0747.16018

Let \(k\) be a commutative ring, \(A\) a Hopf algebra over \(k\) with coproduct \(\delta\), and \(B\) a Hopf subalgebra of \(A\). Letting \(C=A\otimes_ Bk\), \(C\) is a coalgebra and a left \(A\)-module. Under the assumption that \(A\) is a flat right \(B\)-module, the author shows that a left \(A\)-module \(M\) has the form \(A\otimes_ BN\) for some left \(B\)-module \(N\) if, and only if, \(M\) is a left \(C\)-comodule with respect to a map \(\rho: M\to C\otimes M\) such that \(\rho(am)=\delta(a)\cdot\rho(m)\) for \(a\) in \(A\) and \(m\) in \(M\). The proof uses an isomorphism of \(A\otimes_ BM\) onto \(C\otimes M\), the fact that \(\rho\) is the equalizer of the maps \(1\otimes \rho\) and \(\delta\otimes 1\) of \(C\otimes M\) into \(C\otimes C\otimes M\), and an argument by “descent”. If \(E=\text{Hom}_ B(A,k)\) for the left \(B\)- module \(A\), then \(E\) is an algebra and a left \(A\)-module. Assume that \(A\) is a finitely generated, projective left \(B\)-module. By a dual argument, the author shows that a left \(A\)-module \(M\) has the form \(\hbox{Hom}_ B(A,N)\) for some left \(B\)-module \(N\) if, and only if, \(M\) is a right \(E\)- module such that \(a(mf)=\sum(a_{(1)}m)\cdot(a_{(2)}f)\), where \(\delta(a)=\sum a_{(1)}\otimes a_{(2)}\), \(a\) is in \(A\), \(m\) is in \(M\), and \(f\) is in \(E\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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