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

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).

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.

Filtered colimits are exact.

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

## Comments (0)