The Stacks project

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6.32 Closed immersions and (pre)sheaves

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$. In Section 6.21 we have defined functors $i_*$ and $i^{-1}$ such that $i_*$ is right adjoint to $i^{-1}$.

Lemma 6.32.1. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. The stalks of $i_*\mathcal{F}$ are described as follows

\[ i_*\mathcal{F}_ x = \left\{ \begin{matrix} \{ *\} & \text{if} & x \not\in Z \\ \mathcal{F}_ x & \text{if} & x \in Z \end{matrix} \right. \]

where $\{ *\} $ denotes a singleton set. Moreover, $i^{-1}i_* = \text{id}$ on the category of sheaves of sets on $Z$. Moreover, the same holds for abelian sheaves on $Z$, resp. sheaves of algebraic structures on $Z$ where $\{ *\} $ has to be replaced by $0$, resp. a final object of the category of algebraic structures.

Proof. If $x \not\in Z$, then there exist arbitrarily small open neighbourhoods $U$ of $x$ which do not meet $Z$. Because $\mathcal{F}$ is a sheaf we have $\mathcal{F}(i^{-1}(U)) = \{ *\} $ for any such $U$, see Remark 6.7.2. This proves the first case. The second case comes from the fact that for $z \in Z$ any open neighbourhood of $z$ is of the form $Z \cap U$ for some open $U$ of $X$. For the statement that $i^{-1}i_* = \text{id}$ consider the canonical map $i^{-1}i_*\mathcal{F} \to \mathcal{F}$. This is an isomorphism on stalks (see above) and hence an isomorphism.

For sheaves of abelian groups, and sheaves of algebraic structures you argue in the same manner. $\square$

Lemma 6.32.2. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor

\[ i_* : \mathop{\mathit{Sh}}\nolimits (Z) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X) \]

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = \{ *\} $ for all $x \in X \setminus Z$.

Proof. Fully faithfulness follows formally from $i^{-1} i_* = \text{id}$. We have seen that any sheaf in the image of the functor has the property on the stalks mentioned in the lemma. Conversely, suppose that $\mathcal{G}$ has the indicated property. Then it is easy to check that

\[ \mathcal{G} \to i_* i^{-1} \mathcal{G} \]

is an isomorphism on all stalks and hence an isomorphism. $\square$

Lemma 6.32.3. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor

\[ i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X) \]

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

Lemma 6.32.4. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure with final object $0$. The functor

\[ i_* : \mathop{\mathit{Sh}}\nolimits (Z, \mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \]

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

Remark 6.32.5. Let $i : Z \to X$ be a closed immersion of topological spaces as above. Let $x \in X$, $x \not\in Z$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. Then $(i_*\mathcal{F})_ x = \{ * \} $ by Lemma 6.32.1. Hence if $\mathcal{F} = * \amalg *$, where $*$ is the singleton sheaf, then $i_*\mathcal{F}_ x = \{ *\} \not= i_*(*)_ x \amalg i_*(*)_ x$ because the latter is a two point set. According to our conventions in Categories, Section 4.23 this means that the functor $i_*$ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma 4.24.6.

On the other hand, we will see later (see Modules, Lemma 17.6.3) that $i_*$ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on $X$ the sheaf of sections supported in $Z$.

Remark 6.32.6. We have not discussed the relationship between closed immersions and ringed spaces. This is because the notion of a closed immersion of ringed spaces is best discussed in the setting of quasi-coherent sheaves, see Modules, Section 17.13.


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