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.
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}(X). 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
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