Are Biseparable Extensions Frobenius?

Abstract

In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. We show in Section 5 that characterizations of the separable extensions among the Frobenius extensions in [HS, K, K99] are special cases of a result for adjoint functors.

Biseparable extensions are not necessarily Frobenius

“…In [1,Definition 2.4], the notion of a separable module is extended to the concept of biseparable module. When particularizing to ring extensions, [1,Lemma 3.3] says that C ⊆ B is called to be biseparable if one of the following equivalent conditions holds:…”

Formally smooth bimodules

“…The aim of the section is to supplement (and extend) the functorial description of separable bimodules in [4,Corollary 5.8] with the cohomological description of such bimodules. First recall from [16] the following: Recall that a ring morphism A → S is called a separable extension if the product map m S : S ⊗ A S → S has an S-bimodule section.…”

Separable Bimodules and Approximation

“…More results of this type can be found in [5]. In this section we are mainly interested in representation theoretic properties shared by rings related by a bimodule, such as: contravariantly finiteness of the subcategory of modules with finite projective dimension, Finitistic (or finitistic) dimension and representation types, Auslander algebras.…”

“…ness theory. Separable bimodules have been introduced by Sugano [18]; there has been a revived interest recently, see for example [4,5,10] and [11].…”