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Finite depth and Jacobson-Bourbaki correspondence
We introduce a notion of depth three tower of three rings C < B < A with
depth two ring extension A | B recovered when B = C. If A = \End B_C and B | C
is a Frobenius extension, this captures the notion of depth three for a
Frobenius extension in arXiv:math/0107064 and arXiv:math/0108067, such that if
B | C is depth three, then A | C is depth two (a phenomenon of finite depth
subfactors, see arXiv:math/0006057). We provide a similar definition of finite
depth Frobenius extension with embedding theorem utilizing a depth three
subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G
> H > K via the group algebra over a fixed base ring, the depth three condition
is the condition that subgroup K has normal closure K^G contained in H. For a
depth three tower of rings, there is a pre-Galois theory for the ring \End
{}_BA_C and coring (A \o_B A)^C involving Morita context bimodules and left
coideal subrings. This is applied in two sections to a specialization of a
Jacobson-Bourbaki correspondence theorem for augmented rings to depth two
extensions with depth three intermediate division rings.Comment: 26 pp., depth three towers with new section on finite depth, and
correction
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