Depth Two and Infinite Index

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 [6] but extends the main theorem of depth two theory, as for example in [5]. That is, a right depth two extension has right bialgebroid T = (A⊗ 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 .