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 20.45.9. 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 20.45.4. 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 20.45.4 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 20.45.9. 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 20.45.4 we conclude that $i_*C$ is $m$-pseudo-coherent. By induction we conclude that $C$ is $m$-pseudo-coherent. By Cohomology, Lemma 20.45.4 again we conclude that $E$ is $m$-pseudo-coherent. $\square$


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