Modules, comodules, and cotensor products over Frobenius algebras

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A(e)-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M rectangle N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A(e)-module D to N x M, viewed as a left A(e)-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M rectangle N and its derived functors are given by the Hochschild cohomology over A of N x M. (C) 1999 Academic Press.