The Stacks project

Lemma 47.15.2. Let $A$ be a Noetherian ring. Let $K, L \in D_{\textit{Coh}}(A)$ and assume $L$ has finite injective dimension. Then $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is in $D_{\textit{Coh}}(A)$.

Proof. Pick an integer $n$ and consider the distinguished triangle

\[ \tau _{\leq n}K \to K \to \tau _{\geq n + 1}K \to \tau _{\leq n}K[1] \]

see Derived Categories, Remark 13.12.4. Since $L$ has finite injective dimension we see that $R\mathop{\mathrm{Hom}}\nolimits _ A(\tau _{\geq n + 1}K, L)$ has vanishing cohomology in degrees $\geq c - n$ for some constant $c$. Hence, given $i$, we see that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, L) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\tau _{\leq n}K, L)$ is an isomorphism for some $n \gg - i$. By Derived Categories of Schemes, Lemma 36.11.5 applied to $\tau _{\leq n}K$ and $L$ we see conclude that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, L)$ is a finite $A$-module for all $i$. Hence $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is indeed an object of $D_{\textit{Coh}}(A)$. $\square$

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