The Stacks project

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17.6 Closed immersions and abelian sheaves

Recall that we think of an abelian sheaf on a topological space $X$ as a sheaf of $\underline{\mathbf{Z}}_ X$-modules. Thus we may apply any results, definitions for sheaves of modules to abelian sheaves.

Lemma 17.6.1. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Denote $i : Z \to X$ the inclusion map. The functor

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

is exact, fully faithful, with essential image exactly those abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$ is a left inverse to $i_*$.

Proof. Exactness follows from the description of stalks in Sheaves, Lemma 6.32.1 and Lemma 17.3.1. The rest was shown in Sheaves, Lemma 6.32.3. $\square$

Let $\mathcal{F}$ be a sheaf on $X$. There is a canonical subsheaf of $\mathcal{F}$ which consists of exactly those sections whose support is contained in $Z$. Here is the exact statement.

Lemma 17.6.2. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a sheaf on $X$. For $U \subset X$ open set

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

Then $\mathcal{H}_ Z(\mathcal{F})$ is an abelian subsheaf of $\mathcal{F}$. It is the largest abelian subsheaf of $\mathcal{F}$ whose support is contained in $Z$. The construction $\mathcal{F} \mapsto \mathcal{H}_ Z(\mathcal{F})$ is functorial in the abelian sheaf $\mathcal{F}$.

Proof. This follows from Lemma 17.5.2. $\square$

This seems like a good opportunity to show that the functor $i_*$ has a right adjoint on abelian sheaves.

Lemma 17.6.3. Let $i : Z \to X$ be the inclusion of a closed subset into the topological space $X$. Denote1 $i^! : \textit{Ab}(X) \to \textit{Ab}(Z)$ the functor $\mathcal{F} \mapsto i^{-1}\mathcal{H}_ Z(\mathcal{F})$. Then $i^!$ is a right adjoint to $i_*$, in a formula

\[ \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(Z)}(\mathcal{G}, i^!\mathcal{F}). \]

In particular $i_*$ commutes with arbitrary colimits.

Proof. Note that $i_*i^!\mathcal{F} = \mathcal{H}_ Z(\mathcal{F})$. Since $i_*$ is fully faithful we are reduced to showing that

\[ \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})). \]

This follows since the support of the image via any homomorphism of a section of $i_*\mathcal{G}$ is contained in $Z$, see Lemma 17.5.2. $\square$

Remark 17.6.4. In Sheaves, Remark 6.32.5 we showed that $i_*$ as a functor on the categories of sheaves of sets does not have a right adjoint simply because it is not exact. However, it is very close to being true, in fact, the functor $i_*$ is exact on sheaves of pointed sets, sections with support in $Z$ can be defined for sheaves of pointed sets, and $i^!$ makes sense and is a right adjoint to $i_*$.

[1] This is likely nonstandard notation.

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