Skew Hopf algebras, irreducible extensions and the pi-method

To a depth two extension A | B, we associate the dual bialgebroids S := \End {}_BA_B and T := (A \o_B A)^B over the centralizer R=C_A(B). In the set-up where R is a subalgebra of B, which is quite common, two nondegenerate pairings of S and T will define an anti-automorphism \tau of the algebra S. Making use of a two-sided depth two structure, we prove that \tau is an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra. A final section discusses how \tau and the nondegenerate pairings generalize to modules via the pi-method for depth two, and a certain derived mapping of cochain complexes is nullhomotopic.