---- > [!definition] Definition. ([[localization]]) > - Let $R$ be a [[commutative ring|(commutative)]] [[ring]]. >- Let $S \subset R$ be a [[multiplicative subset of a ring|multiplicative subset]]. >- Let $M$ be an $R$-[[module]] > Define an [[equivalence relation]] on the set of pairs $M \times S$ as $(m_{1}, s_{1}) \sim (m_{2}, s_{2}) \iff u(s_{2}m_{1}-s_{1}m_{2})=0\text{ for some }u \in S.$ The class $[(m,s)]$ is denoted $\frac{m}{s}$. We define the **module of fractions** as the set $S ^{-1} M:=\left\{ \frac{m}{s}: (m,s) \in M \times S \right\}$ with addition defined as $\frac{m_{1}}{s_{1}} + \frac{m_{2}}{s_{2}}:= \frac{s_{2}m_{1} + s_{1}m_{2}}{s_{1}s_{2}}$ and $R$-[[module]] structure endowed via $r \cdot \frac{m}{s}:= \frac{rm}{s}\text{ for } r \in R.$ A special case is the **ring of fractions** $S^{-1} R$. > We also have $S ^{-1} R$-[[module]] structure on $S ^{-1} M$, given by $\frac{r}{s_{1}} \cdot \frac{m}{s_{2}} := \frac{rm}{s_{1}s_{2}}. $ The natural [[ring homomorphism]] $\begin{align} R & \xrightarrow{\iota_{S ^{-1} R}} S ^{-1} R \\ r & \mapsto \frac{r}{1} \end{align}$or more generally the [[linear map]] $\begin{align} M & \xrightarrow{\iota_{S ^{-1} R}} S ^{-1} M \\ m & \mapsto \frac{m}{1} \end{align}$ is called the **localization map**. > See also: [[localization functor]]. ^definitiona > [!proposition] Proposition. (Universal property of the ring of fractions) > > Let $U \subset R$ be any subset, $S \subset R$ the [[multiplicative subset of a ring|multiplicative closure]] of $U$. Let $f:R \to B$ be any [[ring homomorphism]] satisfying $f(U) \subset B^{*}$.[^1] Then there exists a unique [[ring homomorphism]] $h:S ^{-1} R \to B$ making the following diagram commute: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGUA9YAWgEYAvgCUQA0uky58hFACZyVWoxZsAQqPEgM2PASJ9SfRfWatEIEQMUwoAc3hFQAMwBOEALZJ5IHBCRklUzYAHWD8HDoAfWAuXkEhAQ1nN09Eb18kA0CVcycQagY6ACMYBgAFSV0ZEBcsWwALHCSQVw9-agzELIYsMDMQKDo4ept87P7QmAAPLDgcOAACAEIF+tEKASA > \begin{tikzcd} > S^{-1}R \arrow[rr, "\exists ! h", dashed] & & B \\ > & R \arrow[lu, "\iota_{S^{-1}R}"] \arrow[ru, "f"'] & > \end{tikzcd} > \end{document} > ``` > [[terminal objects are unique up to a unique isomorphism|As with any object defined by]] a [[universal property]], this characterizes $S ^{-1}R$ up to unique [[isomorphism]]. > > >[!proposition] Corollary. > >There is a natural [[bijection]] (think adjunction) $\text{Hom}_{\mathsf{Ring}}(S ^{-1} R, B) \leftrightarrow \{ \text{ring maps } R \xrightarrow{\varphi} B \text{ satisfying } \varphi(U) \subset B^{*} \}.$ ^proposition > [!basicproperties] Immediate Properties of the ring of fractions $S ^{-1} R$. > 1. Take $\frac{r}{s} \in S ^{-1} R$. Then $\frac{r}{s}=\frac{0}{1} \iff \ex u \in S \text{ s.t. }ur=0$. > 2. $S ^{-1} R=0 \iff \frac{1}{0}=\frac{0}{1} \iff 0 \in S$[^3] > 3. $\ker \iota_{S ^{-1} R}=\{ r \in R : \ex u \in S \text{ s.t. }ur=0 \}$ > 4. $\ker \iota_{S^{-1}R}=\{ 0 \} \iff S$ does not contain [[zero-divisor|zero-divisors]][^4] > 5. $\iota$ is always an [[epimorphism]], but is generally not [[surjection|surjective]].[^5] > > Note that these properties concern the localization of *rings*, not modules. ^properties > [!basicproperties] Localization is 'well-behaved'. > - [[integral algebra|localization and integral closures commute]] > - [[localization commutes with taking radicals]] > - [[localization commutes with intersections, sums, and quotients]] > - [[the localization functor is exact]] > - [[tensor product of modules|Tensoring]]: $S ^{-1} M \otimes_{S ^{-1} R} S ^{-1} N \cong S^{-1}(M \otimes_{R} N)$. ^properties > [!definition] Localization as a quotient. > Another construction satisfying the [[universal property]] above is as follows. Let $S$ be the [[multiplicative subset of a ring|multiplicative closure]] of a subset $U$ of $R$. Consider the $R$-[[algebra]] $R[\{ T_{u} \}_{u \in U}]$. This is a [[polynomial 4|polynomial]] $R$-[[algebra]] with one variable $T_{u}$ for each element of $U$. Now consider the following quotient of the [[free commutative algebra]] on $U$: $R_{U}=\frac{R[\{ T_{u} \}_{u \in U}]}{\langle \underbrace{ \{ uT_{u}-1 \} }_{ =: I_{U} }\rangle }.$ In this sense, $R_{U}$ is obtained by 'adding inverses' to $u \in U$. We show $R_{U} \cong S ^{-1} R$ as [[ring|rings]]. The most important example is for $U=\{ u \}$ a single element. In that special case have $\underbrace{ R_{u} }_{ := \{ u^{n}:n \geq 0 \}^{-1} R } \xrightarrow{\cong} \frac{R[T]}{\langle uT-1 \rangle }.$ ^definition > [!basicexample] The most important examples. > - [ ] todo ($R_{\mathfrak{p}}, R_{f}, \text{Frac }A, etc.$), page 45 ^basic-example > [!proof] Proof of universal property. > **Uniqueness of $h$.** Let $f:R \to B$ be a [[ring homomorphism]] satisfying $f(U) \subset B^{*}$. Let $\frac{r}{s} \in S ^{-1} R$ be arbitrary. We must have[^2] $h\left( \frac{r}{s} \right)=h\left( \frac{r}{1} \frac{1}{s} \right)=h\left( \frac{r}{1} \right)h\left( \frac{s}{1} \right)^{-1}=f(r)f(s)^{-1}.$ **Existence of $h$.** We have to show that $h$, as defined above, is a [[well-defined]] [[ring homomorphism]]. To see well-definition, note that $\begin{align} \frac{r_{1}}{s_{1}}=\frac{r_{2}}{s_{2}} &\iff (\ex t \in s) \ ts_{2}r_{1}=ts_{1}r_{2}\text{ (in }R) \\ & \implies f(t)f(s_{2})f(r_{1})= f(t)f(s_{1}) f(r_{2}). \end{align}$ Now, using that everything is invertible, cancel the $f(t)$s from both sides and rearrange to get $f(r_{1})f(s_{1})^{-1}=f(r_{2})f(s_{2})^{-1}$, as desired. It is evident that $h$ is a [[ring homomorphism]], and so it is indeed well-defined. ^proof > [!specialization] > If $R$ is an [[integral domain]] and $S=R - \{ 0 \}$, then the relation $\sim$ (and in turn the entire construction) is precisely that in [[field of fractions]]. ^specialization ---- #### [^1]: $B^{*}$ denotes the [[unit|group of units]] of $B$. Thus, $f$ is a [[ring homomorphism]] sending elements of $U$ to invertible elements of $B$. [^2]: Note that $f(s)^{-1}$ exists: if $s \in U$ then $f(s) \in B^{*}$ by hypothesis. Else $s=u_{1}\cdots u_{\ell}$ for some $\ell$ and $u_{1},\dots,u_{\ell} \in U$, and $f(s)=f(u_{1}) \cdots f(u_{\ell})$ is a product of units, hence also a unit. [^3]: Slogan: "If you try to divide by zero, you kill everyone". [^4]: Slogan: "If you try to invert a zero-divisor, the localization map won't be injective. If you don't try to invert a zero-divisor, it will be". [^5]: Example: the [[inclusion map]] $\iota:\mathbb{Z} \to \mathbb{Q}$ has $(\beta' \circ \iota)=(\beta'' \circ \iota) \implies (\beta'=\beta'')$ but of course is not [[surjection|surjective]]. - [ ] many verifications ($\sim$ is in fact [[equivalence relation]], addition is well-defined, etc.) ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```