The Stacks project

Lemma 17.3.2. Let $(X, \mathcal{O}_ X)$ be a ringed space.

  1. All limits exist in $\textit{Mod}(\mathcal{O}_ X)$. Limits are the same as the corresponding limits of presheaves of $\mathcal{O}_ X$-modules (i.e., commute with taking sections over opens).

  2. All colimits exist in $\textit{Mod}(\mathcal{O}_ X)$. Colimits are the sheafification of the corresponding colimit in the category of presheaves. Taking colimits commutes with taking stalks.

  3. Filtered colimits are exact.

  4. Finite direct sums are the same as the corresponding finite direct sums of presheaves of $\mathcal{O}_ X$-modules.

Proof. As $\textit{Mod}(\mathcal{O}_ X)$ is abelian (Lemma 17.3.1) it has all finite limits and colimits (Homology, Lemma 12.5.5). Thus the existence of limits and colimits and their description follows from the existence of products and coproducts and their description (see discussion above) and Categories, Lemmas 4.14.11 and 4.14.12. Since sheafification commutes with taking stalks we see that colimits commute with taking stalks. Part (3) signifies that given a system $0 \to \mathcal{F}_ i \to \mathcal{G}_ i \to \mathcal{H}_ i \to 0$ of exact sequences of $\mathcal{O}_ X$-modules over a directed set $I$ the sequence $0 \to \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact as well. Since we can check exactness on stalks (Lemma 17.3.1) this follows from the case of modules which is Algebra, Lemma 10.8.8. We omit the proof of (4). $\square$

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