Are biseparable extensions Frobenius? (Q1611336)

The paper deals with the following interesting problem: Is a finite projective split separable ring extension a Frobenius extension? A positive answer is given in the case of extensions of algebras when one of them is assumed to be separable (Theorem 3.9). Section 1 contains a brief survey on the aforementioned problem. In Section 2 the problem is restated in terms of separable and Frobenius bimodules, accordingly with definitions given by \textit{K. Sugano} [in J. Fac. Sci., Hokkaido Univ., Ser. I 21, 196-208 (1971; Zbl 0236.16003)] and by \textit{F. W. Anderson} and \textit{K. R. Fuller} [in Rings and categories of modules, Graduate Texts in Mathematics, 13. Springer-Verlag (1974; Zbl 0301.16001)] as follows. Let \(M\) be an \(S\)-\(R\)-bimodule, where \(R\) and \(S\) are unitary associative rings, and let \(^*M\) denote the \(R\)-\(S\)-bimodule of all \(S\)-linear homomorphisms from \(M\) to \(S\). The bimodule \(M\) is said to be separable if the canonical evaluation map \(M\otimes_R{^*M}\to S\) is a split epimorphism of \(S\)-bimodules. The bimodule \(M\) is called biseparable when \(M\) is finitely generated and projective as a left \(S\)-module and as a right \(R\)-module, and the bimodules \(M\) and \(M^*\) are separable. Here, \(M^*\) is the \(R\)-\(S\)-bimodule of all \(R\)-linear maps from \(M\) to \(R\). Finally, the bimodule \(M\) is said to be Frobenius if it is finitely generated and projective on both sides and there is an \(R\)-\(S\)-bimodule isomorphism \(^*M\cong M^*\). From this perspective, the original question leads to the more general problem: are biseparable bimodules Frobenius? Section 3 contains some evidences for a positive answer. Thus, for a biseparable ring extension \(R/S\), Proposition 3.6 claims that all \(R\)-modules are \(S\)-relative injective and \(S\)-relative projective, and \(^*R\) and \(R^*\) are generator \(R\)-modules (both duals are now made of \(S\)-linear maps). In Section 4 the authors consider weakening the definition of biseparability in various ways, and find examples of non-Frobenius extensions for each case. A categorical interpretation of the problem, based upon the notions of separable functors and Frobenius functors, is offered in the last section, including some characterizations of Frobenius bimodules given by \textit{F. Castaño Iglesias, C. Năstăsescu} and the reviewer [in Commun. Algebra 27, No. 10, 4879-4900 (1999; Zbl 0949.16038)].