Modules, comodules and cotensor products over Frobenius algebras

Abstract

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 \Box 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 \otimes 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 \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M.