Let $A$ be a Koszul algebra over a field $k$ that is both left and right finite. One can consider the its Koszul complex $X$ as a dg $ A^!$-$A$ bimodule. I want to show:
For all $ i \in \mathbb{Z}$, $ RHom_A(X, X \langle -i \rangle [i]) $ has its homology concentrated in degree $0$.
So far here is what I have been trying: Since $A$ is Koszul, there is a graded projective resolution of $A_0$, call it $P_\bullet$ such that $ AP_i ^i = P_i$. On the other hand as a right $A$ module $X$ is projective resolution of $A_0$. So I try and compute the total Hom space $Hom ^\bullet_A(P_\bullet, A_0\langle -i \rangle [i])$. I am having trouble seeing how this complex is exact everywhere except at $ Hom^0_A(P_\bullet, A_0 \langle -i \rangle [i]) = Hom_A (P_i, A_0 \langle -i \rangle )$.