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\section*{localization of a ring}

\begin{quote}%
This entry is about the general notion of localization of a possible noncommutative ring. For the more restrictive but more traditional notion of [[localization of a commutative ring]] see there.


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{Terminology}{Localization ``at'' and ``away from''}\dotfill \pageref*{Terminology} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_noncommutative_rings}{For noncommutative rings}\dotfill \pageref*{for_noncommutative_rings} \linebreak
\noindent\hyperlink{for_commutative_rings}{For commutative rings}\dotfill \pageref*{for_commutative_rings} \linebreak
\noindent\hyperlink{for_rings}{For $E_k$-rings}\dotfill \pageref*{for_rings} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_a_modality_in_arithmetic_cohesion}{As a modality in arithmetic cohesion}\dotfill \pageref*{as_a_modality_in_arithmetic_cohesion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Given a (possibly noncommutative) unital [[ring]] $R$ there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new ``localized'' ring $S^{-1}R$ equipped with a localization [[homomorphism]] $R\to S^{-1}R$ under which all elements in $S$ are mapped to multiplicatively invertible elements ([[units]]). The latter property must be modified for Cohn localization at multiplicative set of matrices.

We can typically invert elements in a left or right Ore subset $S\subset R$ or much more generally some multiplicative set or matrices ([[Cohn localization]]) etc. There are also some specific localizations like Martindale localizations in ring theory.

\hypertarget{Terminology}{}\subsubsection*{{Localization ``at'' and ``away from''}}\label{Terminology}

The common terminology in algebra is as follows.

For $S$ a set of [[primes]], ``localize at $S$'' means ``invert what is not divisible by $S$''; so for $p$ prime, localizing ``at $p$'' means considering only $p$-[[torsion subgroup|torsion]].

Adjoining inverses $[S^{-1}]$ is pronounced ``localized away from $S$''. Inverting a [[prime]] $p$ is localizing away from $p$, which means ignoring $p$-torsion.

See also lecture notes such as (\hyperlink{Gathmann}{Gathmann}) and see at \emph{[[localization of a space]]} for more discussion of this.

Evidently, this conflicts with more-categorial uses of ``localized''; ``inverting weak equivalences'' is called localization, by obvious analogy, and is written as ``localizing at weak equivalences''. This is confusing! It's also weird: since a ring is a one-object Ab-enriched category with morphisms ``multiply-by'', the localization-of-the-category $R$ ``at $p$'' (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R ``away from p''.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_noncommutative_rings}{}\subsubsection*{{For noncommutative rings}}\label{for_noncommutative_rings}

\begin{defn}
\label{}\hypertarget{}{}
The localization of a [[ring]] $R$ at a [[multiplicative subset]] $S$ is the [[commutative ring]] whose underlying [[set]] is the set of [[equivalence classes]] on $R \times S$ under the [[equivalence relation]]

\begin{displaymath}
(r_1, s_1) \sim (r_2, s_2) 
  \;\;\Leftrightarrow\;\;
  \exists u \in S
  \;
  (r_1 s_2- r_2 s_1) u = 0 \;\in R
  \,.
\end{displaymath}
Write $r s^{-1}$ for the [[equivalence class]] of $(r,s)$. On this set, addition and multiplication is defined by

\begin{displaymath}
r_1 s_1^{-1} + r_2 s_2^{-1} \coloneqq (r_1 s_2 + r_2 s_1) (s_1 s_2)^{-1}
\end{displaymath}
\begin{displaymath}
(r_1 s_1^{-1})(r_2 s_2^{-1}) \coloneqq r_1 r_2 (s_1 s_2)^{-1}
  \,.
\end{displaymath}
\end{defn}
(e.g. [[The Stacks Project|Stacks Project, def. 10.9.1]])

\hypertarget{for_commutative_rings}{}\subsubsection*{{For commutative rings}}\label{for_commutative_rings}

See \emph{[[localization of a commutative ring]]}.

\hypertarget{for_rings}{}\subsubsection*{{For $E_k$-rings}}\label{for_rings}

(\ldots{}) By the lifting property of etale morphisms for $E_k$-rings, see \href{&#233;tale+morphism+of+E-∞+rings#LocalizationOfRings}{here}. (\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_a_modality_in_arithmetic_cohesion}{}\subsubsection*{{As a modality in arithmetic cohesion}}\label{as_a_modality_in_arithmetic_cohesion}

Localization away from a suitably tame ideal may be understood as the [[dR-shape modality]] in the [[cohesion]] of [[E-infinity arithmetic geometry]]:

[[!include arithmetic cohesion -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[localization]],


\item [[localization of abelian groups]]


\item [[localization of a module]]


\item [[localization of a commutative ring]]


\item [[Cohn localization]], [[Ore localization]].


\item [[Bousfield localization of spectra]]

\begin{itemize}%
\item [[localization of a space]] (and of a [[spectrum]])


\item [[p-localization]], [[p-completion]]



\end{itemize}

\item [[completion of a ring]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A classical account of [[localization of commutative rings]] is in section 1 of

\begin{itemize}%
\item [[Dennis Sullivan]], \emph{Geometric topology: localization, periodicity and Galois symmetry}, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by [[Andrew Ranicki]] (\href{http://www.maths.ed.ac.uk/~aar/books/gtop.pdf}{pdf})

\end{itemize}
Further reviews include

\begin{itemize}%
\item Andreas Gathmann, \emph{Localization} ([[GathmannLocalization.pdf:file]])


\item [[Joseph Neisendorfer]] \emph{A Quick Trip through Localization} (\href{https://www.math.rochester.edu/people/faculty/jnei/localization.pdf}{pdf})



\end{itemize}
Discussion of the general concept in [[noncommutative geometry]] is in

\begin{itemize}%
\item [[Zoran ?koda]], \emph{Noncommutative localization in noncommutative geometry}, London Math. Society Lecture Note Series 330 (\href{http://www.maths.ed.ac.uk/~aar/books/nlat.pdf}{pdf}), ed. A. Ranicki; pp. 220--313, \href{http://arxiv.org/abs/math.QA/0403276}{math.QA/0403276}.

\end{itemize}
[[!redirects localizations of a ring]]

[[!redirects localization of rings]] [[!redirects localizations of rings]]



\end{document}