The Stacks project

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17.24 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 17.24.1. 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.

  1. 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 $\mathcal{O}_ X$.

  2. 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 $\mathcal{S}^{-1}\mathcal{O}_ X \to \mathcal{A}$.

  3. For any $x \in X$ we have

    \[ (\mathcal{S}^{-1}\mathcal{O}_ X)_ x = \mathcal{S}_ x^{-1} \mathcal{O}_{X, x}. \]
  4. 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.

  5. 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 17.24.2. 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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