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We study the following question: when is the right adjoint of the forgetful functor from the category of (H,A,C)-Doi-Hopf modules to the category of A-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that C ⊗ A and the smash product A#C∗ are isomorphic as (A,A#C∗)-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case A = H, and this leads to the notion of k-FrobeniusH-module coalgebra. In the special case of Yetter-Drinfel′d modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if H is finite dimensional and unimodular. 0. Introduction Let H be a Hopf algebra with bijective antipode over a commutative ring k. Let A be a (left) H-comodule algebra, and C a (right) H-module coalgebra. Doi [6] introduced the notion of unified Hopf module (we will call such a module a DoiHopf module). This is a k-module, which is at once a right A-module and a left C-comodule, satisfying the compatibility relation (1.7). Several module structures appear as special cases; let us mention Sweedler’s Hopf modules [20], Takeuchi’s relative Hopf modules [21], graded modules, and modules graded by a G-set. It was proved recently [2] that Yetter-Drinfel′d modules are also a special case of Doi-Hopf modules. In [3], induction functors between categories of Doi-Hopf modules are discussed. It turns out that many pairs of adjoint functors are special cases, for example the functor forgetting action or coaction, extension and restriction of scalars and coscalars. In this paper, we focus attention on the functor F from the category of Doi-Hopf modules to the category of right A-modules forgetting the C-coaction. This functor has a right adjoint G = C ⊗ •. A natural question that arises is the following: when is G also a left adjoint of F? In fact, we can view this problem as a Frobenius type problem. It can be shown (cf. [13, Theorem 3.15]) that a ring extension R → S is Frobenius (of the first type) if and only if the induction functor (which is a left adjoint of the restriction Received by the editors May 9, 1995. 1991 Mathematics Subject Classification. Primary 16W30.

Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties