Derived $H$-module endomorphism rings

Let $H$ be a Hopf algebra, $A\mathrm{/}B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^{\cdot }\in D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings $E_A(M^{\cdot })={\text{op}}_{i\in \mathbb{Z}}{\text{Hom}}_{D(A)}(M^{\cdot },M^{\cdot }[i])$ and $E_B(M^{\cdot })={\text{op}}_{i\in \mathbb{Z}}{\text{Hom}}_{D(B)}(M^{\cdot },M^{\cdot }[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^{\cdot })$ is a graded subalgebra of $E_B(M^{\cdot })$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^{\ast }$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.