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\section*{universal localization}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{more_general_definition}{More general definition}\dotfill \pageref*{more_general_definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion \emph{universal localization} or \emph{Cohn localization} of a [[ring]] is a variant of the notion of [[localization of a ring]] which forces not just elements of the ring to become invertible (which one may think of as $1 \times 1$-[[matrices]]) but forces more general [[matrices]] with [[coefficients]] in the ring to become invertible.

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

Let $\Sigma$ be a set of finite square [[matrices]] (of different sizes) over a (typically noncommutative) [[ring]] $R$. Without loss of generality, one assumes that $\Sigma$ is left or right multiplicative. It is left multiplicative if for any matrices $A,B,C$ of right sizes such that $A,B\in\Sigma$ and $C$ fits into matrix $New = \left(\itexarray{ A & 0\\ C & B}\right)$, matrix $New$ is also in $\Sigma$.

We say that a [[homomorphism]] of rings $f: R\to S$ is \emph{$\Sigma$-inverting} if all matrices $f(A)$ over $S$ where $A\in \Sigma$ are invertible in $S$. The Cohn localization of a ring $R$, is a homomorphism of rings $R\to \Sigma^{-1} R$ which is initial in the category of all $\Sigma$-inverting maps (which is the subcategory of coslice category $R/Ring$). In the left hand version, the elements in the localized ring are thought of as solutions of linear equations $A x = b$ where $b$ is a column vector with elements in $R$ and $A\in\Sigma$.

\hypertarget{more_general_definition}{}\subsubsection*{{More general definition}}\label{more_general_definition}

Given a [[ring]] $R$ and a family $S$ of morphisms in the [[category]] $R$[[Mod]] of (say left) [[finitely generated module|finitely generated]] [[projective module|projective]] $R$-[[modules]], we say that a morphism of rings $f:R\to T$ is \textbf{$S$-inverting} if the [[extension of scalars]] from $R$ to $T$ along $f$

\begin{displaymath}
T \otimes_R (-) \colon R Mod \to T Mod
\end{displaymath}
sends all morphisms of $S$ into [[isomorphism]] in the category of left $T$-modules.

P. M. Cohn has shown that there is a universal object $R\to Q_S R$ in the category of $S$-inverting morphisms. The ring $Q_S R$ (and more precisely the universal morphism itself) are called the \textbf{universal localization} or \textbf{Cohn localization} of the ring $R$ at $S$.

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

Cohn localization induces a hereditary torsion theory, i.e. a localization endofunctor on the category of all modules, but it lacks good flatness properties at the level of full module category. However when restricted to the subcategory of finite-dimensional projectives it has all good properties -- it is not any worse than [[Ore localization]].

Universal localization is much used in algebraic K-theory, algebraic L-theory and surgery theory -- see [[Andrew Ranicki]]`s slides in the references at Cohn localization and his papers, specially the series with Amnon Neeman.

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

The existence of the universal localization is exhibited in

\begin{itemize}%
\item P. M. Cohn, \emph{Free rings and their relations}, Academic Press 1971

\end{itemize}
Original articles include

\begin{itemize}%
\item [[P. M. Cohn]], \emph{Inversive localization in noetherian rings}, Communications on Pure and Applied Mathematics \textbf{26}:5-6, pp. 679--691, 1973 \href{http://dx.doi.org/10.1002/cpa.3160260510}{doi}

\end{itemize}
Reviews and lecture notes include

\begin{itemize}%
\item [[V. Retakh]], R. Wilson, \emph{Advanced course on quasideterminants and universal localization}, \href{http://castellet.cat/Publications/quaderns/Quadern41.pdf}{pdf}


\item (NLOC) \emph{Noncommutative localization in algebra and topology}, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002), London Math. Soc. Lecture Notes Series 330 (\href{http://www.maths.ed.ac.uk/~aar/books/nlat.pdf}{pdf}), ed. [[Andrew Ranicki]], Cambridge University Press (2006)


\item [[Z. Škoda]], \emph{Noncommutative localization in noncommutative geometry}, in (NLOC, above) pp. 220--313, \href{http://arxiv.org/abs/math.QA/0403276}{math.QA/0403276}


\item [[Andrew Ranicki]], \emph{Noncommutative localization in algebra and topology}, talk at Knot theory meeting, 2008, slides \href{http://www.maths.ed.ac.uk/~aar/slides/heidel2.pdf}{pdf}; \emph{Noncommutative localization}, [[Pierre Vogel]] 65th birthday conference, Paris, 27 October 2010, \href{http://www.maths.ed.ac.uk/~aar/slides/vogel.pdf}{slides pdf}



\end{itemize}
One can also look at localization with inverses just from one side:

\begin{itemize}%
\item [[P. M. Cohn]], \emph{One-sided localization in rings}, J. Pure Appl. Algebra \textbf{88} (1993), no. 1-3, 37--42 \href{}{}

\end{itemize}
Universal localization of [[group rings]] (and connections to certain noncommutative rational function rings and [[Fox derivative]]s) is discussed in

\begin{itemize}%
\item [[M. Farber]], [[Pierre Vogel]], \emph{The Cohn localization of the free group ring}, Math. Proc. Camb. Phil. Soc. (1992) 111, 433 (\href{http://www.maths.ed.ac.uk/~aar/papers/fv.pdf}{pdf})

\end{itemize}
[[!redirects Cohn localization]] [[!redirects Cohn universal localization]]



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