Relative Hopf modules—Equivalences and freeness criteria

Abstract

Let A be a Hopf algebra over a field k and let B C A be a Hopf subalgebra. The notion of right (A, B)-Hopf modules was introduced by me [l] and the category of those modules was studied to prove that A is a faithfully flat Bmodule, if A is either commutative or cocommutative. Recently Kadford [9] used this notion to know when A is a free (or projective) B-module. In this paper we generalize the notion of relative Hopf modules in two directions, and apply it to obtain many freeness or projectivity criteria for A over B. First, we note that right (A, B)-Hopf modules are defined, if only B is a right coideal subalgebra of A, which means that B is such a subalgebra that d(B)CB @A. Dually, we can define left (T(A), A)-Hopf modules, where w A -+ r(A) is a surjection of coalgebras and left A-modules. In Section 1, we show that the category of right (A, B)-Hopf modules is equivalent to some comodule category, and the categoryof left (a(A), A)-Hopf modules is equivalent to some module category, with a little assumption on flatness. The theorem of Sweedler, which states that the category of right (A, A)-Hopf modules is equivalent to the category of k-vector spaces, follows from this. These equivalences are applied in the commutative case to prove that there is a l-l correspondence B t, 2 between right coideal subalgebras over which A is a faithfully flat module and quotient Hopf algebras over which A is a faithfully coflat left (or equivalently right) comodule. This means that if G is an affine k-group scheme and H C G a closed subgroup scheme, then the dur