The Stacks project

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

  1. 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.

  2. 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.

Proof. This is because tensor product is adjoint to internal hom according to Lemma 18.27.6. See Categories, Lemma 4.24.5. $\square$

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