Sufficiency of 1-dimensional comodule extendibility for Theorem 4.6

Determine whether, for a Hopf algebra H with bijective antipode and a left H-module factor coalgebra C, the extendibility of every 1‑dimensional left C-comodule into a left H-comodule suffices to guarantee that every finite-dimensional left C-comodule embeds as a C-subcomodule in a left H-comodule, as required in Theorem 4.6.

Background

Theorem 4.6 characterizes flatness of H over dominion right coideal subalgebras A via an extendibility property on the corresponding factor coalgebra C: each finite-dimensional C-comodule should embed into an H-comodule. In the classical commutative case H=k[G], the extendibility of 1-dimensional representations already suffices (Białynicki-Birula–Hochschild–Mostow), reflecting observable subgroup theory.

The authors note they cannot derive an analogous strengthening in the general (noncommutative) setting due to the lack of general constructions of 1-dimensional comodules akin to top exterior powers, leaving open whether the 1-dimensional criterion alone suffices.