Centralizers and Inverses to Induction as Equivalence of Categories
Abstract
Description
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 $T = (A ø_B A)^B$. 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 $σ(M)$-categories or its generalizations.
We also show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel 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.
17 pp, additional section discussing Morita equivalence with generalizations applied to the problem in the main body, depth two bimodules, functorial characterizations of left D2 extensions and prebraided commutativity of the centralizer
17 pp, additional section discussing Morita equivalence with generalizations applied to the problem in the main body, depth two bimodules, functorial characterizations of left D2 extensions and prebraided commutativity of the centralizer