Codepth Two and Related Topics
  A depth two extension $A \| B$ is shown to be weak depth two over its double centralizer $V_A(V_A(B))$ if this is separable over $B$. We consider various examples and non-examples of depth one and two properties. Depth two and its relationship to direct and tensor product of algebras as well as cup product of relative Hochschild cochains is examined. Section~6 introduces a notion of codepth two coalgebra homomorphism $g: C \to D$, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms $\End {}^DC^D$ forms a right bialgebroid over the centralizer subalgebra $g^*: D^* \to C^*$ of the dual algebra $C^*$. 
Lars Kadison