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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Ore localization in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > #Contents# * table of contents {:toc} ## Definition ### For a ring If $S\subset R$ is a left [[Ore set]] in a [[monoid]] (or a [[ring]]) $R$, then we call the pair $(j,S^{-1}R)$ where $j:R\to S^{-1}R$ is a morphism of monoids (rings) the __left Ore localization__ of $R$ with respect to $S$ if it is the universal object in the category $C = C(R,S)$ whose objects are the pairs $(f,Y)$ where $f : R \rightarrow Y$ is a morphism of rings from $R$ into a ring $Y$ such that the image $f(S)$ of $S$ consists of units (=multiplicatively invertible elements), and the morphisms $\alpha : (f,Y) \rightarrow (f',Y')$ are maps of rings $\alpha : Y \rightarrow Y'$ such that $\alpha \circ f = f'$. The definition of $C$ makes sense even if $S\subset R$ is not left Ore; the universal object in $C$ may then exist when $S$ is not left Ore, for example this is the case when $S$ is right Ore, while not left Ore. In fact, the universal object is a left Ore localization (i.e. $S$ is left Ore) iff it lies in the full subcategory $C^l$ of $C$ whose objects $(f,Y)$ satisfy two additional conditions: (i) $f(S)^{-1}f(R) = \{(f(s))^{-1}f(r)\,|\, s \in S, r\in R\}$ is a subring in $Y$, (ii) $ker\,f = I_S$. Hence $(j,S^{-1}R)$ is universal in $C^l$, and that characterizes it, but the universality in $C$, although not characteristic, appears to be more useful in practice. For every left Ore set $S\subset R$ in a monoid or ring $R$, the left Ore localization exists and it can be defined as follows. As a set, $S^{-1}R := S\times R/\sim$, where $\sim$ is the following relation of equivalence: \[ (s,r) \sim (s',r') \,\,\Leftrightarrow\,\, (\exists \tilde s\in S \,\,\exists \tilde{r}\in R) \,\, (\tilde{s}s' = \tilde{r}s\,\,and\,\,\tilde{s}r' = \tilde{r}r).\] A class of equivalence of $(s,r)$ is denoted $s^{-1}r$ and called a left fraction. The multiplication is defined by $s_1^{-1}r_1\cdot s_2^{-1}r_2 = (\tilde{s}s_1)^{-1} (\tilde{r}r_2)$ where $\tilde{r} \in R, \tilde{s} \in S$ satisfy $\tilde{r}s_2 = \tilde{s}r_1$ (one should think of this, though it is not yet formally justified at this point, as $s^{-1}\tilde{r} = r_1 s_2^{-1}$, what enables to put inverses one next to another and then the multiplication rule is obvious). If the monoid $R$ is a ring then we can extend the addition to $S^{-1}R$ too. Suppose we are given two fractions with representatives $(s_1,r_1)$ and $(s_2,r_2)$. Then by the left Ore condition we find $\tilde{s} \in S$, $\tilde{r}\in R$ such that $\tilde{s} s_1 = \tilde{r} s_2$. The sum is then defined \[ s_1^{-1} r_1 + s_2^{-1} r_2 \,:=\, (\tilde{s}s_1)^{-1} (\tilde{s}r_1 + \tilde{r}r_2) \] It is a long and at points tricky to work out all the details of this definition. One has to show that $\sim$ is indeed relation of equivalence, that the operations are well defined, and that $S^{-1}R$ is indeed a ring. Even the commutativity of the addition needs work (there is an alternative definition of addition in which $\tilde{s}$ above is not required to be in $S$ but the product $\tilde{s}r_1$ is in $S$; this approach is manifestly commutative but it has some other drawbacks). At the end, one shows that the map $j = j_S : R \rightarrow S^{-1}R$ given by $i(r) = 1^{-1}r$ is a homomorphism of rings, which is 1-1 iff the 2-sided ideal $I_S = \{ n \in R \,|\,\exists s \in S,\, sn = 0\}$ is zero. ### For the category of modules One defines a localization functor which is the extension of scalars $Q^*_S = S^{-1}R\otimes_R - : R-mod\to S^{-1}R-mod$, $M\mapsto S^{-1}R\otimes_R M$. The localization functor is exact ("flat"), has a fully faithful right adjoint, namely the restriction of scalars $Q_{S*}$ and the latter has its own right adjoint (the localization functor is affine). In particular, it realizes $S^{-1}R-mod$ as a reflective subcategory of $R-mod$ and the composition endofunctor $Q_{S*} Q^*_S$ is underlying the corresponding [[idempotent monad]] in $R-mod$. The component of the unit of its adjunction $\eta_R:R\to S^{-1} R$ equals the canonical localization map $j$ and $\eta_M = j\otimes_R\id_M$. ### As a Gabriel localization Given any multiplicative set $S\subset R$, the set of all left ideals $I\subset R$ such that $\forall r\in R$ $\{z\in R| z r\in I\}\cap S\neq 0$ is a [[Gabriel filter]] $\mathcal{F}_S$. If $S$ is left Ore it is sufficient to ask that $I\cap S\neq\emptyset$. The [[Gabriel localization]] functor corresponding to this filter is isomorphic to $Q^*_S$ if $S$ is left Ore. ## Properties Basic property of Ore localization is flatness: $S^{-1}R$ is a flat $R$-bimodule. ## Related concepts * [[Ore domain]] * [[localization of a ring]] * [[Cohn localization]] * [[noncommutative localization]] * [[Gabriel localization]] * [[calculus of fractions]] ## References * [[K. R. Goodearl]], Robert B. Warfield, _An introduction to noncommutative Noetherian rings_, London Math. Soc. Student Texts __16__ (1st ed,), 1989, xviii+303 pp.; or __61__ (2nd ed.), 2004, xxiv+344 pp. * [[Zoran 艩koda]], _Noncommutative localization in noncommutative geometry_, London Math. Society Lecture Note Series 330 ([pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)), ed. A. Ranicki; pp. 220--313, [math.QA/0403276](https://arxiv.org/abs/math.QA/0403276). [[!redirects Ore localizations]] </textarea> </div>  </body> </html>