The Stacks project

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 an abelian sheaf on the topological space $X$. Given a closed subset $Z$, there is a canonical abelian subsheaf of $\mathcal{F}$ which consists of exactly those sections whose support is contained in $Z$. Here is the exact statement.

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

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

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 $Z$. By Lemma 17.6.1 we may (and we do) view $\mathcal{H}_ Z(\mathcal{F})$ as an abelian sheaf on $Z$. In this way we obtain a left exact functor

\[ \textit{Ab}(X) \longrightarrow \textit{Ab}(Z),\quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F}) \text{ viewed as abelian sheaf on }Z \]

All of the statements made above follow directly from Lemma 17.5.2.

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 $X$. The functor $\textit{Ab}(X) \to \textit{Ab}(Z)$, $\mathcal{F} \mapsto \mathcal{H}_ Z(\mathcal{F})$ of Remark 17.6.2 is a right adjoint to $i_* : \textit{Ab}(Z) \to \textit{Ab}(Z)$. In particular $i_*$ commutes with arbitrary colimits.

Proof. We have to show that for any abelian sheaf $\mathcal{F}$ on $X$ and any abelian sheaf $\mathcal{G}$ on $Z$ we have

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

This is clear because after all any section of $i_*\mathcal{G}$ has support in $Z$. Details omitted. $\square$

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 $\mathcal{H}_ Z$ makes sense and is a right adjoint to $i_*$.


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