Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat dimension. 


Q. 1) Prove that $${\bf R}Hom\_R(X,Y)\otimes\_{R}^{\bf L}S\cong{\bf R}Hom\_S(X\otimes\_{R}^{\bf L}S,Y\otimes\_{R}^{\bf L}S)$$.


Q. 2) If $X$ and $Y$ are $R$ modules such that $Tor\_n^R(X,S)=0=Tor\_n^R(Y,S)$ for all $n\geq 1$ then prove that $${\bf R}Hom\_R(X,Y)\otimes\_R^{\bf L}S\cong {\bf R}Hom\_S(X\otimes\_RS, Y\otimes\_RS)$$.


PS: The answers might be straightforward but it would be really helpful if you kindly explain it. Thanks in advance.



Tensoring with complex of finite flat dimension in derived category
Here is a proof of (1) (of course, (2) is a particular case of (1)).


Let me fix Y,S, and let X vary.
The tensor evaluation morphism gives us a morphism


$$
\eta\_X: {\bf R}Hom\_R(X,Y)\otimes\_{R}^{\bf L} S \to {\bf R}Hom\_R(X,Y\otimes\_{R}^{\bf L} S )
$$


Now, because $S$ has finite flat dimension over $R$, and $Y$ is bounded below, both of the functors
$${\bf R}Hom\_R(-,Y)\otimes\_{R}^{\bf L} S $$
and 
$${\bf R}Hom\_R(-,Y\otimes\_{R}^{\bf L} S )$$
are way-out left functors (in the sense of Residues and duality, Chapter I.7).


Since $\eta\_R$ is clearly an isomorphism, we deduce by the lemma on way out functors (Dual of residues and duality, Proposition I.7.1) that $\eta\_X$ is an isomorphism for all $X \in D^{-}\_f(R)$.


Finally, note that by the (derived) hom tensor-adjunction:
$$
{\bf R}Hom\_S(X \otimes\_{R}^{\bf L} S,Y\otimes\_{R}^{\bf L} S ) \cong
{\bf R}Hom\_R(X,{\bf R}Hom\_S(S,Y\otimes\_{R}^{\bf L} S )) = 
{\bf R}Hom\_R(X,Y \otimes\_{R}^{\bf L} S ))
$$


