Lemma 75.8.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ such that the support of $H^ i(E)$ is proper over $Y$ for all $i$. Then $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$.

## 75.8 Derived category of coherent modules

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. In this case the category $\textit{Coh}(\mathcal{O}_ X) \subset \textit{Mod}(\mathcal{O}_ X)$ of coherent $\mathcal{O}_ X$-modules is a weak Serre subcategory, see Homology, Section 12.10 and Cohomology of Spaces, Lemma 69.12.3. Denote

the subcategory of complexes whose cohomology sheaves are coherent, see Derived Categories, Section 13.17. Thus we obtain a canonical functor

see Derived Categories, Equation (13.17.1.1).

**Proof.**
Consider the spectral sequence

see Derived Categories, Lemma 13.21.3. By assumption and Lemma 75.7.10 the sheaves $R^ pf_*H^ q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $E \in D_{\textit{Coh}}(\mathcal{O}_ Y)$. Boundedness from below is trivial. Boundedness from above follows from Cohomology of Spaces, Lemma 69.8.1 or from Lemma 75.6.1. $\square$

Lemma 75.8.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ such that the support of $H^ i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\textit{Coh}}(\mathcal{O}_ Y)$.

**Proof.**
The proof is the same as the proof of Lemma 75.8.1. You can also deduce it from Lemma 75.8.1 by considering what the exact functor $Rf_*$ does to the distinguished triangles $\tau _{\leq a}E \to E \to \tau _{\geq a + 1}E \to \tau _{\leq a}E[1]$.
$\square$

Lemma 75.8.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. 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.**
We can check whether an object of $D(\mathcal{O}_ X)$ is in $D_{\textit{Coh}}(\mathcal{O}_ X)$ étale locally on $X$, see Cohomology of Spaces, Lemma 69.12.2. Hence this lemma follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.11.5.
$\square$

Lemma 75.8.4. Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$. For $L$ in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, the $A$-modules $\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L)$ are finite.

**Proof.**
Recall that

see Cohomology on Sites, Lemma 21.35.1 and Cohomology on Sites, Section 21.14. Thus the result follows from Lemmas 75.8.3 and 75.8.2. $\square$

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