The Stacks project

Lemma 36.11.5. Let $X$ be a locally Noetherian scheme. If $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. It suffices to prove this when $X$ is the spectrum of a Noetherian ring $A$. By Lemma 36.10.3 we see that $K$ is pseudo-coherent. Then we can use Lemma 36.10.8 to translate the problem into the following algebra problem: for $L \in D^+_{\textit{Coh}}(A)$ and $K$ in $D^-_{\textit{Coh}}(A)$, then $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is in $D^+_{\textit{Coh}}(A)$. Since $L$ is bounded below and $K$ is bounded below there is a convergent spectral sequence

\[ \mathop{\mathrm{Ext}}\nolimits ^ p_ A(K, H^ q(L)) \Rightarrow \text{Ext}^{p + q}_ A(K, L) \]

and there are convergent spectral sequences

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(H^{-j}(K), H^ q(L)) \Rightarrow \text{Ext}^{i + j}_ A(K, H^ q(L)) \]

See Injectives, Remarks 19.13.9 and 19.13.11. This finishes the proof as the modules $\mathop{\mathrm{Ext}}\nolimits ^ p_ A(M, N)$ are finite for finite $A$-modules $M$, $N$ by Algebra, Lemma 10.71.9. $\square$

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