Depth three towers of rings and groups

Abstract

Depth three and finite depth are notions known for subfactors via diagrams and Frobenius extensions of rings via centralizers in endomorphism towers. From the point of view of depth two ring extensions, we provide a clear definition of depth three for a tower of three rings C < B < A. If A = B_C and B | C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension. For example we provide an algebraic proof that if B | C is depth three, then A | C is depth two. 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 an interesting algebraic theory from the point of view of Galois correspondence theory for the ring {}_BA_C and coring (A \o_B A)^C with respect to the centralizers A^B and A^C involving Morita context bimodules, nondegenerate pairings and right coideal subrings.