Given any topological space $X$, a closed subset $Z \subset X$, and an abelian sheaf $\mathcal{F}$ you can take the subsheaf of sections whose support is contained in $Z$. If $X$ is a scheme, $Z$ a closed subscheme, and $\mathcal{F}$ a quasi-coherent module there is a variant where you take sections which are scheme theoretically supported on $Z$. However, in the scheme setting you have to be careful because the resulting $\mathcal{O}_ X$-module may not be quasi-coherent.
Lemma 28.24.1. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be an open subscheme. The following are equivalent:
$U$ is retrocompact in $X$,
$U$ is quasi-compact,
$U$ is a finite union of affine opens, and
there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ such that $X \setminus U = V(\mathcal{I})$ (set theoretically).
Proof. The equivalence of (1), (2), and (3) follows from Lemma 28.2.3. Assume (1), (2), (3). Let $T = X \setminus U$. By Schemes, Lemma 26.12.4 there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subscheme structure on $T$. Note that $\mathcal{J}|_ U = \mathcal{O}_ U$ which is an $\mathcal{O}_ U$-modules of finite type. By Lemma 28.22.2 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 (4). Conversely, if $\mathcal{I}$ is as in (4) and $W = \mathop{\mathrm{Spec}}(R) \subset X$ is an affine open, then $\mathcal{I}|_ W = \widetilde{I}$ for some finitely generated ideal $I \subset R$, see Lemma 28.16.1. It follows that $U \cap W = \mathop{\mathrm{Spec}}(R) \setminus V(I)$ is quasi-compact, see Algebra, Lemma 10.29.1. Hence $U \subset X$ is retrocompact by Lemma 28.2.6. $\square$
Lemma 28.24.2. Let $X$ be a scheme. 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 open $U \subset X$ Assume $\mathcal{I}$ is of finite type. Then
$\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules,
on any affine open $U \subset X$ we have $\mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\} $, and
$\mathcal{F}'_ x = \{ s \in \mathcal{F}_ x \mid \mathcal{I}_ x s = 0\} $.
\[ \mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\} \]
Proof. It is clear that the rule defining $\mathcal{F}'$ gives a subsheaf of $\mathcal{F}$ (the sheaf condition is easy to verify). Hence we may work locally on $X$ to verify the other statements. In other words we may assume that $X = \mathop{\mathrm{Spec}}(A)$, $\mathcal{F} = \widetilde{M}$ and $\mathcal{I} = \widetilde{I}$. It is clear that in this case $\mathcal{F}'(U) = \{ x \in M \mid Ix = 0\} =: M'$ because $\widetilde{I}$ is generated by its global sections $I$ which proves (2). To show $\mathcal{F}'$ is quasi-coherent it suffices to show that for every $f \in A$ we have $\{ x \in M_ f \mid I_ f x = 0\} = (M')_ f$. Write $I = (g_1, \ldots , g_ t)$, which is possible because $\mathcal{I}$ is of finite type, see Lemma 28.16.1. If $x = y/f^ n$ and $I_ fx = 0$, then that means that for every $i$ there exists an $m \geq 0$ such that $f^ mg_ ix = 0$. We may choose one $m$ which works for all $i$ (and this is where we use that $I$ is finitely generated). Then we see that $f^ mx \in M'$ and $x/f^ n = f^ mx/f^{n + m}$ in $(M')_ f$ as desired. The proof of (3) is similar and omitted. $\square$
Definition 28.24.3. Let $X$ be a scheme. 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 28.24.2 above is called the subsheaf of sections annihilated by $\mathcal{I}$.
Lemma 28.24.4. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. 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 so that Lemma 28.24.2 applies to $\mathcal{I}$ and $f_*\mathcal{F}$.) $\square$
For an abelian sheaf on a topological space we have discussed the subsheaf of sections with support in a closed subset in Modules, Remark 17.6.2. For quasi-coherent modules this submodule isn't always a quasi-coherent module, but if the closed subset has a retrocompact complement, then it is.
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 $U \subset X$ If $X \setminus Z$ is a retrocompact open of $X$, then
for an affine open $U \subset X$ there exist a finitely generated ideal $I \subset \mathcal{O}_ X(U)$ such that $Z \cap U = V(I)$,
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\} $,
$\mathcal{F}'$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules.
\[ \mathcal{F}'(U) = \{ s \in \mathcal{F}(U) \mid \text{the support of }s\text{ is contained in }Z \cap U\} \]
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
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
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
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. $\square$
Definition 28.24.6. Let $X$ be a scheme. Let $T \subset X$ be a closed subset whose complement is retrocompact in $X$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The quasi-coherent subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma 28.24.5 is called the subsheaf of sections supported on $T$.
Lemma 28.24.7. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $Z \subset Y$ be a closed subset such that $Y \setminus Z$ is retrocompact in $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported in $f^{-1}Z$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported in $Z$.
Proof. Omitted. (Hint: First show that $X \setminus f^{-1}Z$ is retrocompact in $X$ as $Y \setminus Z$ is retrocompact in $Y$. Hence Lemma 28.24.5 applies to $f^{-1}Z$ and $\mathcal{F}$. As $f$ is quasi-compact and quasi-separated we see that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma 28.24.5 applies to $Z$ and $f_*\mathcal{F}$. Finally, match the sheaves directly.) $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #120 by Pieter Belmans on
Comment #121 by Johan on