Complete H-equivariant Morita class representatives via H0 lifts

Establish that if H is a finite-dimensional Hopf algebra with the dual Chevalley property, H0 its coradical, and X0 a complete set of exact H0-equivariant Morita equivalence class representatives, then X := {A: λ^{-1}(H0 A) is isomorphic to an element in X0} is a complete set of exact H-equivariant Morita equivalence class representatives.

Background

The paper studies module categories over tensor categories arising from finite-dimensional Hopf algebras by classifying H-equivariant Morita equivalence classes of AM-exact H-comodule algebras. A key tool is a lifting method that builds Loewy-graded H-comodule algebras from data determined at the coradical H0.

The conjecture proposes that a classification at the level of H can be reduced to degree-zero data at H0: namely, that choosing representatives for exact H0-equivariant Morita equivalence classes and then lifting them via the Loewy filtration yields a complete set of representatives for exact H-equivariant Morita equivalence classes. The paper proves this conjecture along the first step (for Loewy-graded cases), but the full assertion remains conjectural.