Lemma 17.16.5. Let (X, \mathcal{O}_ X) be a ringed space. For any \mathcal{O}_ X-module \mathcal{F} the functor
commutes with arbitrary colimits.
Lemma 17.16.5. Let (X, \mathcal{O}_ X) be a ringed space. For any \mathcal{O}_ X-module \mathcal{F} the functor
commutes with arbitrary colimits.
Proof. Let I be a preordered set and let \{ \mathcal{G}_ i\} be a system over I. Set \mathcal{G} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i. Recall that \mathcal{G} is the sheaf associated to the presheaf \mathcal{G}' : U \mapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i(U), see Sheaves, Section 6.29. By Lemma 17.16.2 the tensor product \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} is the sheafification of the presheaf
where the equality sign is Algebra, Lemma 10.12.9. Hence the lemma follows from the description of colimits in \textit{Mod}(\mathcal{O}_ X), see Lemma 17.3.2. \square
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Comment #6216 by Yuto Masamura on
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