The Stacks project

Lemma 36.12.5. Let $i : Z \to X$ be a morphism of ringed spaces such that $i$ is a closed immersion of underlying topological spaces and such that $i_*\mathcal{O}_ Z$ is pseudo-coherent as an $\mathcal{O}_ X$-module. Let $E \in D(\mathcal{O}_ Z)$. Then $E$ is $m$-pseudo-coherent if and only if $Ri_*E$ is $m$-pseudo-coherent.

Proof. Throughout this proof we will use that $i_*$ is an exact functor, and hence that $Ri_* = i_*$, see Modules, Lemma 17.6.1.

Assume $E$ is $m$-pseudo-coherent. Let $x \in X$. We will find a neighbourhood of $x$ such that $i_*E$ is $m$-pseudo-coherent on it. If $x \not\in Z$ then this is clear. Thus we may assume $x \in Z$. We will use that $U \cap Z$ for $x \in U \subset X$ open form a fundamental system of neighbourhoods of $x$ in $Z$. After shrinking $X$ we may assume $E$ is bounded above. We will argue by induction on the largest integer $p$ such that $H^ p(E)$ is nonzero. If $p < m$, then there is nothing to prove. If $p \geq m$, then $H^ p(E)$ is an $\mathcal{O}_ Z$-module of finite type, see Cohomology, Lemma 20.47.9. Thus we may choose, after shrinking $X$, a map $\mathcal{O}_ Z^{\oplus n}[-p] \to E$ which induces a surjection $\mathcal{O}_ Z^{\oplus n} \to H^ p(E)$. Choose a distinguished triangle

\[ \mathcal{O}_ Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_ Z^{\oplus n}[-p + 1] \]

We see that $H^ j(C) = 0$ for $j \geq p$ and that $C$ is $m$-pseudo-coherent by Cohomology, Lemma 20.47.4. By induction we see that $i_*C$ is $m$-pseudo-coherent on $X$. Since $i_*\mathcal{O}_ Z$ is $m$-pseudo-coherent on $X$ as well, we conclude from the distinguished triangle

\[ i_*\mathcal{O}_ Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_ Z^{\oplus n}[-p + 1] \]

and Cohomology, Lemma 20.47.4 that $i_*E$ is $m$-pseudo-coherent.

Assume that $i_*E$ is $m$-pseudo-coherent. Let $z \in Z$. We will find a neighbourhood of $z$ such that $E$ is $m$-pseudo-coherent on it. We will use that $U \cap Z$ for $z \in U \subset X$ open form a fundamental system of neighbourhoods of $z$ in $Z$. After shrinking $X$ we may assume $i_*E$ and hence $E$ is bounded above. We will argue by induction on the largest integer $p$ such that $H^ p(E)$ is nonzero. If $p < m$, then there is nothing to prove. If $p \geq m$, then $H^ p(i_*E) = i_*H^ p(E)$ is an $\mathcal{O}_ X$-module of finite type, see Cohomology, Lemma 20.47.9. Choose a complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ Z$-modules representing $E$. We may choose, after shrinking $X$, a map $\alpha : \mathcal{O}_ X^{\oplus n}[-p] \to i_*\mathcal{E}^\bullet $ which induces a surjection $\mathcal{O}_ X^{\oplus n} \to i_*H^ p(\mathcal{E}^\bullet )$. By adjunction we find a map $\alpha : \mathcal{O}_ Z^{\oplus n}[-p] \to \mathcal{E}^\bullet $ which induces a surjection $\mathcal{O}_ Z^{\oplus n} \to H^ p(\mathcal{E}^\bullet )$. Choose a distinguished triangle

\[ \mathcal{O}_ Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_ Z^{\oplus n}[-p + 1] \]

We see that $H^ j(C) = 0$ for $j \geq p$. From the distinguished triangle

\[ i_*\mathcal{O}_ Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_ Z^{\oplus n}[-p + 1] \]

the fact that $i_*\mathcal{O}_ Z$ is pseudo-coherent and Cohomology, Lemma 20.47.4 we conclude that $i_*C$ is $m$-pseudo-coherent. By induction we conclude that $C$ is $m$-pseudo-coherent. By Cohomology, Lemma 20.47.4 again we conclude that $E$ is $m$-pseudo-coherent. $\square$


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 09VA. Beware of the difference between the letter 'O' and the digit '0'.