Modules, comodules and cotensor products over Frobenius algebras

Lowell Abrams

16D90; 16E30

Department of Mathematics, Hill Center, Rutgers University, New
Brunswick, NJ 08903


   This is a corrected version, with significant expository
improvements.

   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.

   This paper has been submitted to the Journal of Algebra, and
copyright may be transferred.