Faithful flatness of Hopf algebras (Q1340006)

The aim of this paper is to study the flatness and faithful flatness of Hopf algebras over right coideal subalgebras. Let \(A\) be a Hopf algebra with bijective antipode and \(B\) a right coideal subalgebra of \(A\). Then \(A\) is faithfully flat as a left \(B\)-module if and only if \(A\) is flat as a left \(B\)-module and \(B\) is a simple object in the category \(M^ A_ B\), and also if and only if \(A\) is a projective generator as a left \(B\)- module. The authors prove that a commutative Hopf algebra is a flat module over every right coideal subalgebra. Hence a commutative Hopf algebra \(A\) is faithfully flat as a \(B\)-module if and only if \(B\) is a simple object in \(M^ A_ B\) where \(B\) is a right coideal subalgebra of \(A\). Note that a Hopf subalgebra is necessarily a right coideal subalgebra. Applying the results in this paper, an important theorem of \textit{M. Takeuchi} [Manuscr. Math. 7, 251-270 (1972; Zbl 0238.16011)] is proved easily: a commutative Hopf algebra is a faithfully flat module, or more strongly a projective generator over every Hopf subalgebra.