Anchor maps and stable modules in depth two

An algebra extension A | B is right depth two if its tensor-square A\otimes_B A is in the Dress category Add A as A-B-bimodules. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu math.QA/9905192, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids S = End {}_BA_B and T = End {}_AA\otimes_BA_A over the centralizer R = C_A(B) are the modules {}_SR and R_T studied in math.RA/0505004, math.RA/0408155 and math.GR/0409346, which provide information about the bialgebroids and the extension (cf. math.QA/0409106). The anchor maps for the Hopf algebroids in math.KT/0105105 and math.QA/0508411 reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in math.QA/0508638. We sketch a theory of stable $A$-modules and their endomorphism rings and generalize the smash product decomposition in Prop. 1.1, (L. Kadison, Hopf Algebroid and H-separable extensions, Proc. A.M.S. 131 (2003), 2993-3002) to any A-module. We observe that Schneider's coGalois theory (Isr.J.Math 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.