The Stacks project

Theorem 19.7.4. The category of sheaves of abelian groups on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition 12.27.5.

Proof. Let $\mathcal{G}_ i$, $i \in I$ be a set of abelian sheaves such that every subsheaf of every $\mathbf{Z}_ X^\# $ occurs as one of the $\mathcal{G}_ i$. Apply Lemma 19.7.2 to this collection to get an ordinal $\beta $. We claim that for any sheaf of abelian groups $\mathcal{F}$ the map $\mathcal{F} \to J_\beta (\mathcal{F})$ is an injection of $\mathcal{F}$ into an injective. Note that by construction the assignment $\mathcal{F} \mapsto \big (\mathcal{F} \to J_\beta (\mathcal{F})\big )$ is indeed functorial.

The proof of the claim comes from the fact that by Lemma 19.7.3 it suffices to extend any morphism $\gamma : \mathcal{G} \to J_\beta (\mathcal{F})$ from a subsheaf $\mathcal{G}$ of some $\mathbf{Z}_ X^\# $ to all of $\mathbf{Z}_ X^\# $. Then by Lemma 19.7.2 the map $\gamma $ lifts into $J_\alpha (\mathcal{F})$ for some $\alpha < \beta $. Finally, we apply Lemma 19.7.1 to get the desired extension of $\gamma $ to a morphism into $J_{\alpha + 1}(\mathcal{F}) \to J_\beta (\mathcal{F})$. $\square$

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