The Stacks project

Lemma 17.27.2. Let $X$ be a topological space. Let $\mathcal{O}_ X$ be a presheaf of rings. Let $\mathcal{S} \subset \mathcal{O}_ X$ be a pre-sheaf of sets contained in $\mathcal{O}_ X$. Suppose that for every open $U \subset X$ the set $\mathcal{S}(U) \subset \mathcal{O}_ X(U)$ is a multiplicative subset. For any presheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we have

\[ \mathcal{S}^{-1}\mathcal{F} = \mathcal{S}^{-1}\mathcal{O}_ X \otimes _{p, \mathcal{O}_ X} \mathcal{F} \]

(see Sheaves, Section 6.6 for notation) and if $\mathcal{F}$ and $\mathcal{O}_ X$ are sheaves then

\[ (\mathcal{S}^{-1}\mathcal{F})^\# = (\mathcal{S}^{-1}\mathcal{O}_ X)^\# \otimes _{\mathcal{O}_ X} \mathcal{F} \]

(see Sheaves, Section 6.20 for notation).

Proof. Omitted. $\square$

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