localization of a module

Let $R$ be a commutative ring and $M$ an $R$-module. Let $S\subset R$ be a non-empty multiplicative set. Form the Cartesian product $M\times S$, and define a binary relation $\sim$ on $M\times S$ as follows:

$(m_1,s_1)\sim (m_2,s_2)$ if and only if there is some $t\in S$ such that $t(s_2{m}_1-s_1{m}_2)=0$

Proposition 1.

$\sim$ on $M\mathrm{\times }S$ is an equivalence relation.

Proof.

Clearly $(m,s)\sim (m,s)$ as $t(sm-sm)=0$ for any $t\in S$, where $S\ne \mathrm{\varnothing }$. Also, $(m_1,s_1)\sim (m_2,s_2)$ implies that $(m_2,s_2)\sim (m_1,s_1)$, since $t(s_2{m}_1-s_1{m}_2)=0$ implies that $t(s_1{m}_2-s_2{m}_1)=0$. Finally, given $(m_1,s_1)\sim (m_2,s_2)$ and $(m_2,s_2)\sim (m_3,s_3)$, we are led to two equations $t(s_2{m}_1-s_1{m}_2)=0$ and $u(s_3{m}_2-s_2{m}_3)=0$ for some $t,u\in S$. Expanding and rearranging these, then multiplying the first equation by $u{s}_3$ and the second by $t{s}_1$, we get $tu{s}_2(s_3{m}_1-s_1{m}_3)=0$. Since $tu{s}_2\in S$, $(m_1,s_1)\sim (m_3,s_3)$ as required. ∎

Let $M_S$ be the set of equivalence classes in $M\times S$ under $\sim$. For each $(m,s)\in M\times S$, write

$$[(m,s)]\text{ or more commonly }\frac{m}{s}$$

the equivalence class in $M_S$ containing $(m,s)$. Next,

• define a binary operation $+$ on $M_S$ as follows:

  $$\frac{m_1}{s_1}+\frac{m_2}{s_2}:=\frac{s_2{m}_1+s_1{m}_2}{s_1{s}_2}.$$

• define a function $\cdot :R_S\times M_S\to M_S$ as follows:

  $$\frac{r}{s}\cdot \frac{m}{t}:=\frac{rm}{st}$$

  where $R_S$ is the localization of $R$ over $S$.

Proposition 2.

$M_S$ together with $\mathrm{+}$ and $\mathrm{\cdot }$ defined above is a unital module over $R_S$.

Proof.

That $+$ and $\cdot$ are well-defined is based on the following: if $(m_1,s_1)\sim (m_2,s_2)$, then

$$\frac{m}{s}+\frac{m_1}{s_1}=\frac{m}{s}+\frac{m_2}{s_2},\frac{m_1}{s_1}+\frac{m}{s}=\frac{m_2}{s_2}+\frac{m}{s},\text{and}\mathit{\hspace{1em}}\frac{r}{s}\cdot \frac{m_1}{s_1}=\frac{r}{s}\cdot \frac{m_2}{s_2},$$

which are clear by Proposition $1$. Furthermore $+$ is commutative and associative and that $\cdot$ distributes over $+$ on both sides, which are all properties inherited from $M$. Next, ${\displaystyle \frac{0}{s}}$ is the additive identity in $M_S$ and ${\displaystyle \frac{-m}{s}}\in M_S$ is the additive inverse of ${\displaystyle \frac{m}{s}}$. So $M_S$ is a module over $R_S$. Finally, since $(mt,st)\sim (m,s)$ for any $t\in S$, ${\displaystyle \frac{t}{t}}\cdot {\displaystyle \frac{m}{s}}={\displaystyle \frac{m}{s}}$ so that $M_S$ is unital. ∎

Definition. $M_S$, as an $R_S$-module, is called the localization of $M$ at $S$. $M_S$ is also written $S^{-1}M$.

Remarks.

• The notion of the localization of a module generalizes that of a ring in the sense that $R_S$ is the localization of $R$ at $S$ as an $R_S$-module.

• If $S=R-𝔭$, where $𝔭$ is a prime ideal in $R$, then $M_S$ is usually written $M_{𝔭}$.

Title	localization of a module
Canonical name	LocalizationOfAModule
Date of creation	2013-03-22 17:26:59
Last modified on	2013-03-22 17:26:59
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 13B30