Frobenius functors: Applications

We investigate functors between abelian categories having isomorphic left and right adjoint functors (these functors are called Frobenius Functors). They are characterized for categories of modules and categories of comodules. We give some applications in coalgebras and Hopf modules. In particular, we introduce the notion of Frobenius homomorphism of coalgebras. The set of isomorphism classes of Frobenius functors between quite general Grothendieck categories is endowed with an abelian group structure. This gives a functorial notion of Grothendieck group which behaves satisfactorily.