An exercise on derived category in Weibel's book: commuting $\mathbf{R}\mathrm{Hom}_R$ with $\otimes^{\mathbf{L}}_R$ in a "weird" way My question is on the following exercise in Weibel's book: Exercise 10.8.3.
Exercise 10.8.3: Let $R$ be a commutative ring and $C$ a bounded complex of finite $\mathrm{Tor}$-dimension over $R$. Show that there is a natural isomorphism in the derived category $\mathsf{D}(R-\mathrm{Mod})$:
$$
\mathbf{R}\mathrm{Hom}_R(A,B) \otimes^{\mathbf{L}}_R C \cong \mathbf{R}\mathrm{Hom}_R(A,B \otimes^{\mathbf{L}}_R C ).
$$
How I got stuck: I feel that the general way to prove such problems is in two steps:

*

*Degenerate the desired isomorphism to the zeroth degree and prove the zeroth degree version.


*Use the composition theorem for derived functors (i.e. $\mathbf{R}(GF) \cong \mathbf{R}G \circ \mathbf{R}F$) to upgrade to the derived version.
At least for most exercises and theorem I have met, the thing works smoothly:

Example 1: To show
$$
\mathbf{R}\mathrm{Hom}_R(A,\mathbf{R}\mathrm{Hom}_R(B,C)) \cong \mathbf{R}\mathrm{Hom}_R(A \otimes^{\mathbf{L}}_R B, C )  \quad\quad (\star),
$$
we first degenerate it to the zeroth degree as
$$
\mathrm{Hom}_R(A, \mathrm{Hom}_R(B,C)) \cong \mathrm{Hom}_R(A \otimes_R B, C).
$$
This is the clear adjoint isomorphism. Then we view both sides as composed functors in $B$. Then if $A$ is a projective complex and $C$ is an injective complex, we check the conditions of the composition theorem of derived functors hold. So we upgrade both sides to the derived version. Then we take projective resolution of $A$ and injective resolution of $C$ in general, we have the final isomorphism $(\star)$.


Example 2: We can also prove the Shapiro's lemma in group cohomology by this method, by first degenerate to the canonical isomorphism
$$
\mathrm{Hom}_G(\mathbb{Z}, \mathrm{Ind}^G_H(-)) \cong \mathrm{Hom}_H(\mathbb{Z}, -)
$$
and upgrade it to
$$
\mathbf{R}\mathrm{Hom}_G(\mathbb{Z}, \mathrm{Ind}^G_H(-)) \cong \mathbf{R}\mathrm{Hom}_H(\mathbb{Z}, -),
$$
by further noticing $\mathrm{Ind}^G_H(-)$ is an exact functor. Then taking the $n$-th cohomology, we get the Shapiro's lemma.

But my experience fails when facing the exercise above, since the isomorphism on the zeroth degree level, i.e. commuting tensor product with Hom in such a weird (for me) way is quite out of reach for me. Yet I have no idea how to tackle in another approach.

Similar questions arises when I'm trying to solve the following exercise in Weibel's book:
Exercise 10.8.4: Let $f: R \rightarrow S$ is a flat ring homomorphism, and $f^{\ast}$ is the functor sending an $R$-module $A$ to the $S$-module $A \otimes_{R} S$. Now suppose $A$ is quasi-isomorphic to a bounded above complex of finitely generated projective modules. Show that we have a natural isomorphism for every $B$ in $\mathsf{D}^{+}(R-\mathrm{Mod})$:
$$
\mathbf{L}f^{\ast} \mathbf{R}\mathrm{Hom}_R(A,B) \rightarrow \mathbf{R}\mathrm{Hom}_S(\mathbf{L}f^{\ast}A, \mathbf{L}f^{\ast} B).
$$
And similar to the previous exercise, I also got stuck at the very first beginning, feeling confused on the degenerated version.
Or maybe my two-step approach has some hidden troubles that I haven't realized? Or is it quite limited in the application of derived category?
Quite frustrated! :(
Thank you all for helping, commenting or answering, or even reading such a long post! :)
 A: The underived version of this isomorphism
$$\text{Hom}(A, B) \otimes C \cong \text{Hom}(A, B \otimes C)$$
is false without quite strong hypotheses; you can think of it as trying to commute a limit, namely $\text{Hom}(A, -)$, with a colimit, namely $(-) \otimes C$, and limits and colimits generally don't commute without further hypotheses. For this setup, since the hypotheses are on $C$ I'll give hypotheses on $C$: with no hypotheses on $A, B$ we need $C$ to be finitely generated projective, or equivalently finitely presented flat. I suppose "bounded complex of finite Tor-dimension" is a derived analogue of this but I'm not familiar with the details.
Other sets of hypotheses that make this true are

*

*$A$ is finitely generated projective

*$A$ is finitely presented and $C$ is flat

*$A$ is projective and $C$ is finitely presented.

See this MO answer for some discussion.
For a counterexample when these hypotheses are dropped take $R = \mathbb{Z}, A = \mathbb{Q}, B = \mathbb{Z}, C = \mathbb{Q}$. Note that neither $A$ nor $C$ are finitely presented.
A: As Qiaochu Yuan shows in his answer, the statement in this exercise is false without extra hypotheses. In fact, it turns out that Weibel knew this.
In a list of corrections to the 1995 edition, he includes a correction to this exercise, so that the statement is that there is a natural homomorphism in $D(R)$, which is an isomorphism if either $C$
is a complex of finitely generated projective $R$-modules or else $A$ is quasi-isomorphic to a bounded below chain complex of finitely generated projective $R$-modules.
