Centralizers and Inverses to Induction as Equivalence of Categories
| dc.creator | Kadison, Lars | |
| dc.date | 2005-04-30 | |
| dc.date | 2005-06-02 | |
| dc.date.accessioned | 2026-07-07T05:19:33Z | |
| dc.date.available | 2026-07-07T05:19:33Z | |
| dc.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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0505004 | |
| dc.identifier | http://arxiv.org/abs/math/0505004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75054 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Category Theory | |
| dc.subject | 16D90, 18E05, 20D25, 22D30 | |
| dc.title | Centralizers and Inverses to Induction as Equivalence of Categories | |
| dc.type | text |