17.27 Localizing sheaves of rings
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 presheaf 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, see Algebra, Definition 10.9.1. In this case we can consider the presheaf of rings
\[ \mathcal{S}^{-1}\mathcal{O}_ X : U \longmapsto \mathcal{S}(U)^{-1}\mathcal{O}_ X(U). \]
The restriction mapping sends the section $f/s$, $f \in \mathcal{O}_ X(U)$, $s \in \mathcal{S}(U)$ to $(f|_ V)/(s|_ V)$ if $V \subset U$ are opens of $X$.
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
\[ (\mathcal{S}^{-1}\mathcal{O}_ X)^\# _ x = \mathcal{S}_ x^{-1} \mathcal{O}_{X, x}. \]
Proof.
Omitted.
$\square$
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 presheaf 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. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_ X$-modules. In this case we can consider the presheaf of $\mathcal{S}^{-1}\mathcal{O}_ X$-modules
\[ \mathcal{S}^{-1}\mathcal{F} : U \longmapsto \mathcal{S}(U)^{-1}\mathcal{F}(U). \]
The restriction mapping sends the section $t/s$, $t \in \mathcal{F}(U)$, $s \in \mathcal{S}(U)$ to $(t|_ V)/(s|_ V)$ if $V \subset U$ are opens of $X$.
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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