The Stacks project

Lemma 47.13.3. Let $\varphi : A \to B$ be a ring homomorphism. For $K$ in $D(A)$ we have

\[ \varphi _*R\mathop{\mathrm{Hom}}\nolimits (B, K) = R\mathop{\mathrm{Hom}}\nolimits _ A(B, K) \]

where $\varphi _* : D(B) \to D(A)$ is restriction. In particular $R^ q\mathop{\mathrm{Hom}}\nolimits (B, K) = \mathop{\mathrm{Ext}}\nolimits _ A^ q(B, K)$.

Proof. Choose a K-injective complex $I^\bullet $ representing $K$. Then $R\mathop{\mathrm{Hom}}\nolimits (B, K)$ is represented by the complex $\mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )$ of $B$-modules. Since this complex, as a complex of $A$-modules, represents $R\mathop{\mathrm{Hom}}\nolimits _ A(B, K)$ we see that the lemma is true. $\square$

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