The Stacks project

68.14 Sections with support in a closed subset

This section is the analogue of Properties, Section 28.24.

Lemma 68.14.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U \subset X$ be an open subspace. The following are equivalent:

  1. $U \to X$ is quasi-compact,

  2. $U$ is quasi-compact, and

  3. there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ such that $|X| \setminus |U| = |V(\mathcal{I})|$.

Proof. Let $W$ be an affine scheme and let $\varphi : W \to X$ be a surjective étale morphism, see Properties of Spaces, Lemma 64.6.3. If (1) holds, then $\varphi ^{-1}(U) \to W$ is quasi-compact, hence $\varphi ^{-1}(U)$ is quasi-compact, hence $U$ is quasi-compact (as $|\varphi ^{-1}(U)| \to |U|$ is surjective). If (2) holds, then $\varphi ^{-1}(U)$ is quasi-compact because $\varphi $ is quasi-compact since $X$ is quasi-separated (Morphisms of Spaces, Lemma 65.8.10). Hence $\varphi ^{-1}(U) \to W$ is a quasi-compact morphism of schemes by Properties, Lemma 28.24.1. It follows that $U \to X$ is quasi-compact by Morphisms of Spaces, Lemma 65.8.8. Thus (1) and (2) are equivalent.

Assume (1) and (2). By Properties of Spaces, Lemma 64.12.3 there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subspace structure on $|X| \setminus |U|$. Note that $\mathcal{J}|_ U = \mathcal{O}_ U$ which is an $\mathcal{O}_ U$-modules of finite type. As $U$ is quasi-compact it follows from Lemma 68.9.2 that there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_ U = \mathcal{J}|_ U$. Then $|X| \setminus |U| = |V(\mathcal{I})|$ and we obtain (3). Conversely, if $\mathcal{I}$ is as in (3), then $\varphi ^{-1}(U) \subset W$ is a quasi-compact open by the lemma for schemes (Properties, Lemma 28.24.1) applied to $\varphi ^{-1}\mathcal{I}$ on $W$. Thus (2) holds. $\square$

Lemma 68.14.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. 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 object $U$ of $X_{\acute{e}tale}$ the module

\[ \mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\} \]

Assume $\mathcal{I}$ is of finite type. Then

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

  2. for affine $U$ in $X_{\acute{e}tale}$ we have $\mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\} $, and

  3. $\mathcal{F}'_ x = \{ s \in \mathcal{F}_ x \mid \mathcal{I}_ x s = 0\} $.

Proof. It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$. Hence we may work étale locally on $X$ to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 28.24.2. $\square$

Definition 68.14.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma 68.14.2 above is called the subsheaf of sections annihilated by $\mathcal{I}$.

Lemma 68.14.4. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\mathcal{I}\mathcal{O}_ X$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections annihilated by $\mathcal{I}$.

Proof. Omitted. Hint: The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent (Morphisms of Spaces, Lemma 65.11.2) so that Lemma 68.14.2 applies to $\mathcal{I}$ and $f_*\mathcal{F}$. $\square$

Next we come to the sheaf of sections supported in a closed subset. Again this isn't always a quasi-coherent sheaf, but if the complement of the closed is “retrocompact” in the given algebraic space, then it is.

Lemma 68.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset and let $U \subset X$ be the open subspace such that $T \amalg |U| = |X|$. 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 object $\varphi : W \to X$ of $X_{\acute{e}tale}$ the module

\[ \mathcal{F}'(W) = \{ s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi |^{-1}(T)\} \]

If $U \to X$ is quasi-compact, then

  1. for $W$ affine there exist a finitely generated ideal $I \subset \mathcal{O}_ X(W)$ such that $|\varphi |^{-1}(T) = V(I)$,

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

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

Proof. It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$. Hence we may work étale locally on $X$ to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 28.24.5. $\square$

Definition 68.14.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset whose complement corresponds to an open subspace $U \subset X$ with quasi-compact inclusion morphism $U \to X$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The quasi-coherent subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma 68.14.5 above is called the subsheaf of sections supported on $T$.

Lemma 68.14.7. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $T \subset |Y|$ be a closed subset. Assume $|Y| \setminus T$ corresponds to an open subspace $V \subset Y$ such that $V \to Y$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported on $|f|^{-1}T$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported on $T$.

Proof. Omitted. Hints: $|X| \setminus |f|^{-1}T$ is the support of the open subspace $U = f^{-1}V \subset X$. Since $V \to Y$ is quasi-compact, so is $U \to X$ (by base change). The assumption that $f$ is quasi-compact and quasi-separated implies that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma 68.14.5 applies to $T$ and $f_*\mathcal{F}$ as well as to $|f|^{-1}T$ and $\mathcal{F}$. The equality of the given quasi-coherent modules is immediate from the definitions. $\square$


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