Depth Three Towers of Rings and Groups

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 = EndBC and B |C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension. 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 contained in H. For a depth three tower of rings, there is the beginning of an interesting algebraic theory for the ring End BAC and coring (A⊗BA) C with respect to the centralizers VA(B) and VA(C) involving Morita context bimodules, nondegenerate pairings and comodules.