Centralizers and Inverses to Induction as Equivalence of Categories

Given a ring homomorphism B → A, consider its centralizer R = A, bimodule endomorphism ring S = EndBAB and sub-tensor-square ring T = (A ⊗B A) B . Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A |B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which replaces the key module AAe for an Azumaya algebra A. After establishing these characterizations, we characterize left D2 extensions in terms of the module TR, and ask whether a weak generator condition on RT might characterize left D2 extensions as well, possibly a problem in σ(M)-categories or its generalizations. We also show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel and pre-braided commutative. For example, its normality yields a Hopf subalgebra analogue of a factoid for subgroups and their centralizers, and a special case of a conjecture that D2 Hopf subalgebras are normal.