H-equivariant morita equivalences of loewy-graded comodule algebras

Let H be a coradically graded Hopf algebra. For every Loewy-graded exact H -comodule algebra A=⊕ n≥ 0 A(n) A = ⊕ n ≥ 0 A ( n ) and H₀ H 0 -equivariant Morita equivalence A(0) H₀ X A ( 0 ) ≃ H 0 X , there exists a Loewy-graded H -comodule algebra B (isomorphic to X in degree zero) realizing an H -equivariant Morita equivalence A H B A ≃ H B . In addition, if every exact H₀ H 0 -comodule algebra is H₀ H 0 -equivariant Morita equivalent to a coideal subalgebra of H₀ H 0 , then every Loewy-graded exact H -comodule algebra is H -equivariant Morita equivalent to a coideal subalgebra of H . We also discuss Loewy-graded H -comodule algebras with H₀=KP H 0 = K P , the Kac-Paljutkin Hopf algebra.