The structure of Frobenius and separable algebras
by
Bogdan Ion
Princeton University

We shall study Frobenius algebras and separable algebras from a unifying point of view. The tool will be what we have called the Frobenius-separability equation: R¹²R²³=R²³R¹³=R¹³R¹², where A is an algebra over a commutative ring k and R in A \otimesA. We can impose two different normalisation conditions for R: the Frobenius, respectively the separability normalisation condition. If R is a solution of the Frobenius-separability equation, we shall construct a k-algebra A(R), which is a Frobenius algebra, respectively a separable algebra, depending on the normalisation conditions we impose. The main result of this paper is the structure of these two fundamental types of algebras: if k is a field and n a positive integer, then any n-dimensional Frobenius algebra, respectively separable algebra, is isomorphic with an A(n, R). This provides a description with generators and relations of these algebras. On the route, a new caracterisation of Frobenius extension is given: a ring extension B subset A is a Frobenius extension if and only if A is a "coalgebra" over B. Finaly, we shall introduce the category _AFS^A of FS-objects and we shall refine a result of L. Abrams by showing that, over a Frobenius algebra A, there exists an isomorphism of categories _AFS^A =~ _AM =~ M^A.

Date received: December 2, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-36.