Tensoring with complex of finite flat dimension in derived category

Question

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.

Answer 1

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: $$ {\bf R}Hom_S(X \otimes_{R}^{\bf L} S,Y\otimes_{R}^{\bf L} S ) \cong {\bf R}Hom_R(X,{\bf R}Hom_S(S,Y\otimes_{R}^{\bf L} S )) = {\bf R}Hom_R(X,Y \otimes_{R}^{\bf L} S )) $$