Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Let H be a Hopf algebra over a field k and A a right coideal subalgebra of H , that is, A is a subalgebra satisfying ∆(A) ⊂ A⊗H where ∆ is the comultiplication in H . In case when H is finitely generated commutative, the right coideal subalgebras are intimately related to the homogeneous spaces for the corresponding group scheme. The purpose of this paper is to extend the class of pairsA,H for whichH is proved to be either projective or flat as a module over A. As is known the faithful flatness over Hopf subalgebras may be lacking in general. Examples given by Schauenburg [26] use some extremely big Hopf algebras coming from a universal construction of [31]. Positive results can be expected therefore only under some finiteness assumptions. A Hopf algebra is called residually finite dimensional [19] if its ideals of finite codimension have zero intersection. Many important classes of Hopf algebras are residually finite dimensional. Among them are the finitely generated commutative Hopf algebras, the universal enveloping algebras of finite dimensional Lie algebras, and also Hopf algebras related to quantum groups. We say that a ring R has semilocal localizations with respect to a central subring Z if for each maximal ideal m of Z the localization Rm of R at the multiplicatively closed set Z r m is a semilocal ring whose Jacobson radical contains mRm. For instance, this property is satisfied for any ring module-finite over a central subring. For each ring R let MR and RM denote the categories of right and left R-modules, respectively.