Anchor Maps and Stable Modules in Depth Two

An algebra extension A |B is right depth two if its tensor-square A⊗B A is in the Dress category Add BAA. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of twosided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, 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 = EndBAB and T = End AA ⊗B AA over the centralizer R = CA(B) are the modules SR and RT studied in [11, 15, 8], which provide information about the bialgebroids and the extension [9]. The anchor maps for the Hopf algebroids in [17, 10] reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in [20]. We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in [7, Prop. 1.1] to any A-module. We observe that Schneider’s coGalois theory in [21] provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra.