Centralizers and Inverses to Induction as Equivalence of Categories

Given a ring homomorphism $B \to A$, consider its centralizer $R = A^B$, bimodule endomorphism ring $S = \End {}_BA_B$ and sub-tensor-square ring $T = (A \o_B A)^B$. Nonassociative tensoring by the cyclic modules $R_T$ 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 $R_T$ or ${}_SR$ are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is $R_T$ a progenerator, which replaces the key module $A_{A^e}$ for an Azumaya algebra $A$. After establishing these characterizations, we characterize left D2 extensions in terms of the module $T_R$, and ask whether a weak generator condition on $R_T$ might characterize left D2 extensions as well, possibly a problem in $\sigma(M)$-categories or its generalizations. We also show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as 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.