Separability and the twisted Frobenius bimodule (Q1968748)

The author recalls Frobenius extensions \(S\to A\), denoted \(A/S\), and more generally \(\beta\)-Frobenius extensions, where \(_SS_S\) is replaced by the bimodule \(_\beta S_S\), the \(\beta\) indicating the left \(S\)-module structure on \(S\) given by the automorphism \(\beta\) of the ring \(S\). It is known that if \(K\) is a Hopf subalgebra of a finite-dimensional Hopf algebra \(H\), then \(H/K\) is a (free) \(\beta\)-extension where \(\beta\) is constructed from the Nakayama automorphisms of \(K\) and \(H\). This paper studies a generalization of \(\beta\)-Frobenius extensions to Frobenius bimodules with two-sided twisting and its relation to separability (when separability implies Frobenius is an open question). He proves an endomorphism ring theorem, and its converse for certain twisted Frobenius bimodules. He discusses a duality between separable extensions and split extensions. He characterizes the twisted Frobenius bimodules that are separable. He shows that if \(A/S\) is \(\beta\)-Frobenius, then it is split (respectively separable) if and only if \(\text{End}(A_S)/A\) is separable (respectively split). He discusses when projective separability implies Frobenius. In a final section, he gives examples of free separable \(\beta\)-Frobenius extensions of finite rank which are not Frobenius extensions in the usual sense. One example is the Taft Hopf algebra \(H\) of dimension \(n^2\) with \(K\) the Hopf subalgebra which is a group algebra of the cyclic group of order \(n\). This is a split \(\beta\)-Frobenius extension, but \(\beta\) is not an \(H\)-inner automorphism of \(K\). The other example is a certain algebra \(A\) of 4 by 4 matrices with \(S\) the subalgebra of diagonal matrices. The \(\beta\) involved is not \(A\)-inner. Both \(_SA\) and \(A_S\) are free of rank two. \(A/S\) here is split but not separable. If \(E=\text{End}_S(A_S)\), then \(E/A\) is separable but not split.