Existence of a coflat but not faithfully coflat factor coalgebra

Determine whether there exists a left H-module factor coalgebra C of a Hopf algebra H with bijective antipode such that H is left or right coflat over C but not left or right faithfully coflat, respectively; construct an explicit example if such a coalgebra exists, or prove that coflatness always implies faithful coflatness for left H-module factor coalgebras.

Background

The paper studies Takeuchi’s correspondence between right coideal subalgebras A⊂H and left H‑module factor coalgebras C=H/I for Hopf algebras H with bijective antipode, focusing on weakening faithful (co)flatness assumptions. In classical and several structured cases, coflatness of H over C is known to imply faithful coflatness: for example, when C is a Hopf factor algebra (Doi) or when H has cocommutative coradical (Masuoka).

The authors prove new sufficient conditions (e.g., Theorem 7.4) where coflatness together with flatness on the opposite side forces faithful coflatness. Despite these results, the general existence of a factor coalgebra C for which H is coflat but not faithfully coflat remains unresolved.