Quasi-Frobenius functors with application to corings

Müller generalized in [12] the notion of a Frobenius extension to left (right) quasi-Frobenius extension and proved the endomorphism ring theorem for these extensions. Recently, Guo observed in [9] that for a ring homomorphism φ : R → S, the restriction of scalars functor has to induction functor S ⊗R − : RM → SM as right ”quasi” adjoint if and only if φ is a left quasi-Frobenius extension. In this paper we shall give a further generalization of the notion of quasi-Frobenius extension in a more general setting from the viewpoint of an adjoint triple of functors. Using the definition of quasi-strongly adjoint pair for module categories given by K. Morita [11], we introduce the notion of quasi-Frobenius triple of functors for Grothendieck categories. A triple of functors (L,F,R), where F : A → B has a left adjoint L : B → A and a right adjoint R : B → A is said to be quasi-Frobenius whenever the functors L andR are similar in sense functorial. In this case, the functor F is called quasi-Frobenius functor. F is a Frobenius functor in the case it has the same right and left adjoint, i.e., L ∼= R (cf. [5]). Clearly, the class of quasi-Frobenius functors include to the class of Frobenius functors. First we study basic properties of quasi-Frobenius functors for Grothendieck categories. This concept generalizes la notion of left quasi-Frobenius pair of functors given in Guo [9]. In Section 2 we give an easy and natural proof of the characterization of quasi-Frobenius functors between module categories. In particular, a bijective correspondence between quasi-Frobenius triple of functors is presented (in fact, a duality). Another interesting case is given by graded rings and modules and this is considered in Section 3. The notion of Frobenius extension for coalgebras over fields was