Modules, comodules and cotensor products over Frobenius algebras
Lowell Abrams
16D90; 16E30
Department of Mathematics, Hill Center, Rutgers University, New
Brunswick, NJ 08903
labrams@math.rutgers.edu
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 comodules over A, relative to this
coproduct, is isomorphic to the category of right 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 right A^e-module D to M \otimes N,
viewed as a right A^e-module. Some of the properties of D are
investigated, and some sample calculations are given. Finally, we show
that when A is commutative or semisimple, the cotensor product M \Box
N and its derived functors are given by the Hochschild cohomology over
A of M \otimes N.
This paper has been submitted to the Journal of Algebra, and
copyright may be transferred.