Derived equivalence, recollements under $H$-Galois extensions

In this paper, assume that $H$ is a Hopf algebra and $A/B$ is an $H$-Galois extension. Firstly, by introducing the concept of an $H$-stable tilting complex $T_{\bullet }$ over $B$, we show that $T_{\bullet }{\otimes }_{B}A$ is a tilting complex over $A$ and a derived equivalence between two $H$-module algebras can be extended to smash product algebras under some conditions. Then we observe that $0\to {\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathcal{D}}^b(B)}(T_{\bullet })\to {\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathcal{D}}^b(A)}(T_{\bullet }{\otimes }_{B}A)$ is an $H$-Galois Frobenius extension if $A/B$ is an $H$-Galois Frobenius extension. Finally, for any perfect recollement of derived categories of $H$-module algebras, we apply the above results to construct a perfect recollement of derived categories of their smash product algebras and generalize it to $n$-recollements.