The Stacks project

Remark 13.24.3. Let $\textit{Mod}(\mathcal{O}_ X)$ be the category of $\mathcal{O}_ X$-modules on a ringed space $(X, \mathcal{O}_ X)$ (or more generally on a ringed site). We will see later that $\textit{Mod}(\mathcal{O}_ X)$ has enough injectives and in fact functorial injective embeddings, see Injectives, Theorem 19.8.4. Note that the proof of Lemma 13.23.4 does not apply to $\textit{Mod}(\mathcal{O}_ X)$. But the proof of Lemma 13.24.1 does apply to $\textit{Mod}(\mathcal{O}_ X)$. Thus we obtain

\[ j : K^{+}(\textit{Mod}(\mathcal{O}_ X)) \longrightarrow K^{+}(\mathcal{I}) \]

which is a resolution functor where $\mathcal{I}$ is the additive category of injective $\mathcal{O}_ X$-modules. This argument also works in the following cases:

  1. The category $\text{Mod}_ R$ of $R$-modules over a ring $R$.

  2. The category $\textit{PMod}(\mathcal{O})$ of presheaves of $\mathcal{O}$-modules on a site endowed with a presheaf of rings.

  3. The category $\textit{Mod}(\mathcal{O})$ of sheaves of $\mathcal{O}$-modules on a ringed site.

  4. Add more here as needed.

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