Lemma 17.27.1. 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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