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
\[ D_{\textit{Coh}}(\mathcal{O}_ X) \subset D(\mathcal{O}_ X) \]
the subcategory of complexes whose cohomology sheaves are coherent, see Derived Categories, Section 13.17. Thus we obtain a canonical functor
75.8.0.1
\begin{equation} \label{spaces-perfect-equation-compare-coherent} D(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_ X) \end{equation}
see Derived Categories, Equation (13.17.1.1).
Proof. Consider the spectral sequence
\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \]
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
\[ \mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L) = H^ n(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L)) = H^ n(\mathop{\mathrm{Spec}}(A), Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L)) \]
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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