Infinite index subalgebras of depth two
  An algebra extension $A \| B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory math.RA/0108067 but extends the main theorem of depth two theory, as for example in math.RA/0107064. That is, a right depth two extension has right bialgebroid T = (A \otimes_B A)^B$ over its centralizer R = C_A(B). The main theorem: an extension A | B is right depth two and right balanced if and only if A | B is T-Galois wrt. left projective, right R-bialgebroid T. 
Lars Kadison