Lemma 47.13.10. Let $R \to A$ be a surjective ring map whose kernel $I$ is an invertible $R$-module. The functor $R\mathop{\mathrm{Hom}}\nolimits (A, -) : D(R) \to D(A)$ is isomorphic to $K \mapsto K \otimes _ R^\mathbf {L} N[-1]$ where $N$ is inverse of the invertible $A$-module $I \otimes _ R A$.
Proof. Since $A$ has the finite projective resolution
\[ 0 \to I \to R \to A \to 0 \]
we see that $A$ is a perfect $R$-module. By Lemma 47.13.9 it suffices to prove that $R\mathop{\mathrm{Hom}}\nolimits (A, R)$ is represented by $N[-1]$ in $D(A)$. This means $R\mathop{\mathrm{Hom}}\nolimits (A, R)$ has a unique nonzero cohomology module, namely $N$ in degree $1$. As $\text{Mod}_ A \to \text{Mod}_ R$ is fully faithful it suffice to prove this after applying the restriction functor $i_* : D(A) \to D(R)$. By Lemma 47.13.3 we have
\[ i_*R\mathop{\mathrm{Hom}}\nolimits (A, R) = R\mathop{\mathrm{Hom}}\nolimits _ R(A, R) \]
Using the finite projective resolution above we find that the latter is represented by the complex $R \to I^{\otimes -1}$ with $R$ in degree $0$. The map $R \to I^{\otimes -1}$ is injective and the cokernel is $N$. $\square$
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