Separable extensions for crossed products over monoidal Hom-Hopf algebras

Let (H, alpha) be a Frobenius monoidal Hom-Hopf algebra, and (A, beta) an (H*, alpha*(-1))-Hom-Hopf Galois extension of (A(H), beta vertical bar H-A). We prove that the separability of the Hom-algebra extension A/A(H) is equivalent to the existence of a trace one element omega is an element of C-A#H(A) that centralizes A. As applications, we obtain the differentiated conditions for the extension A#H-sigma/A to be separable, and deduce a Doi's result of Hom-type.