Lemma 18.27.7. Tensor product and colimits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings.

For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

\[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} \]commutes with arbitrary colimits.

Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings. For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

\[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \]commutes with arbitrary colimits.

## Comments (0)