The Stacks project

Remark 19.9.5. Let $\mathcal{C}$ be a site. Note that $\textit{Ab}(\mathcal{C})$ has enough injectives, see Theorem 19.7.4. (In the case that $\mathcal{C}$ has enough points this is straightforward because $p_*I$ is an injective sheaf if $I$ is an injective $\mathbf{Z}$-module and $p$ is a point.) Also, $\textit{Ab}(\mathcal{C})$ has a cogenerator (details omitted). Hence Lemma 19.9.2 proves that we have a fully faithful, exact embedding $\mathcal{A} \to \mathcal{B}$ where $\mathcal{B}$ has a cogenerator and enough injectives. We can apply this to $\mathcal{A}^{opp}$ and we get a fully faithful exact functor $i : \mathcal{A} \to \mathcal{D} = \mathcal{B}^{opp}$ where $\mathcal{D}$ has enough projectives and a generator. Hence $\mathcal{D}$ has a projective generator $P$. Set $R = \mathop{Mor}\nolimits _\mathcal {D}(P, P)$. Then

\[ \mathcal{A} \longrightarrow \text{Mod}_ R, \quad X \longmapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(P, X). \]

One can check this is a fully faithful, exact functor. In other words, one retrieves the Freyd-Mitchell theorem mentioned in Remark 19.9.3 above.


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