Lemma 17.6.1 (01AX)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.6: Closed immersions and abelian sheaves
Lemma 17.6.1 (cite)

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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. $\square$

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