Abstract
Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = EndBABand sub-tensor-square ring T = (A ⊗BA)B. Nonassociative tensoring by the cyclic modules RTorSR 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 RTorSR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RTa progenerator, which takes the place of the key module AAefor an Azumaya algebra A. In addition, we characterize left D2 extensions in terms of the module TR, and show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, the notion of normality yields a version for Hopf subalgebras of the fact that normal subgroups have normal centralizers, and yields a special case of a conjecture that D2 Hopf subalgebras are normal.
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