Lemma 17.24.1. Let $X$ be a topological space and 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.

There is a map of presheaves of rings $\mathcal{O}_ X \to \mathcal{S}^{-1}\mathcal{O}_ X$ such that every local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{O}_ X$.

For any homomorphism of presheaves of rings $\mathcal{O}_ X \to \mathcal{A}$ such that each local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{A}$ there exists a unique factorization $\mathcal{S}^{-1}\mathcal{O}_ X \to \mathcal{A}$.

For any $x \in X$ we have

\[ (\mathcal{S}^{-1}\mathcal{O}_ X)_ x = \mathcal{S}_ x^{-1} \mathcal{O}_{X, x}. \]The sheafification $(\mathcal{S}^{-1}\mathcal{O}_ X)^\# $ is a sheaf of rings with a map of sheaves of rings $(\mathcal{O}_ X)^\# \to (\mathcal{S}^{-1}\mathcal{O}_ X)^\# $ which is universal for maps of $(\mathcal{O}_ X)^\# $ into sheaves of rings such that each local section of $\mathcal{S}$ maps to an invertible section.

For any $x \in X$ we have

\[ (\mathcal{S}^{-1}\mathcal{O}_ X)^\# _ x = \mathcal{S}_ x^{-1} \mathcal{O}_{X, x}. \]

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