The Stacks project

73.8 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 67.12.3. 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

73.8.0.1
\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).

Lemma 73.8.1. 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 $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$.

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 73.7.10 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 67.8.1 or from Lemma 73.6.1. $\square$

Lemma 73.8.2. 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 73.8.1. You can also deduce it from Lemma 73.8.1 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 73.8.3. 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 $D_{\textit{Coh}}(\mathcal{O}_ X)$ ├ętale locally on $X$, see Cohomology of Spaces, Lemma 67.12.2. Hence this lemma follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.11.5. $\square$

Lemma 73.8.4. 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 21.34.1 and Cohomology on Sites, Section 21.14. Thus the result follows from Lemmas 73.8.3 and 73.8.2. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08GI. Beware of the difference between the letter 'O' and the digit '0'.