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 12.24.5.

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 $J_{\alpha + 1}(\mathcal{F}) \to J_\beta (\mathcal{F})$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01DP. Beware of the difference between the letter 'O' and the digit '0'.