The Stacks project

Lemma 28.24.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Consider the sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}'$ which associates to every open $U \subset X$

\[ \mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \text{the support of }s\text{ is contained in }Z \cap U\} \]

If $X \setminus Z$ is a retrocompact open of $X$, then

  1. for an affine open $U \subset X$ there exist a finitely generated ideal $I \subset \mathcal{O}_ X(U)$ such that $Z \cap U = V(I)$,

  2. for $U$ and $I$ as in (1) we have $\mathcal{F}'(U) = \{ x \in \mathcal{F}(U) \mid I^ nx = 0 \text{ for some } n\} $,

  3. $\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules.

Proof. Part (1) is Algebra, Lemma 10.29.1. Let $U = \mathop{\mathrm{Spec}}(A)$ and $I$ be as in (1). Then $\mathcal{F}|_ U$ is the quasi-coherent sheaf associated to some $A$-module $M$. We have

\[ \mathcal{F}'(U) = \{ x \in M \mid x = 0\text{ in }M_\mathfrak p \text{ for all }\mathfrak p \not\in Z\} . \]

by Modules, Definition 17.5.1. Thus $x \in \mathcal{F}'(U)$ if and only if $V(\text{Ann}(x)) \subset V(I)$, see Algebra, Lemma 10.40.7. Since $I$ is finitely generated this is equivalent to $I^ n x = 0$ for some $n$. This proves (2).

Proof of (3). Observe that given $U \subset X$ open there is an exact sequence

\[ 0 \to \mathcal{F}'(U) \to \mathcal{F}(U) \to \mathcal{F}(U \setminus Z) \]

If we denote $j : X \setminus Z \to X$ the inclusion morphism, then we observe that $\mathcal{F}(U \setminus Z)$ is the sections of the module $j_*(\mathcal{F}|_{X \setminus Z})$ over $U$. Thus we have an exact sequence

\[ 0 \to \mathcal{F}' \to \mathcal{F} \to j_*(\mathcal{F}|_{X \setminus Z}) \]

The restriction $\mathcal{F}|_{X \setminus Z}$ is quasi-coherent. Hence $j_*(\mathcal{F}|_{X \setminus Z})$ is quasi-coherent by Schemes, Lemma 26.24.1 and our assumption that $j$ is quasi-compact (any open immersion is separated). Hence $\mathcal{F}'$ is quasi-coherent as a kernel of a map of quasi-coherent modules, see Schemes, Section 26.24. $\square$

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