nLab
universal localization

Definition

Let $\Sigma$ be a mulitplicative family of matrices over a (typically noncommutative) ring $R$.

We say that a map of rings $f:R\to S$ is $\Sigma$-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 $R/\mathrm{Ring}$).

More general definition

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

$$T{\otimes }_R(-):R\mathrm{Mod}\to T\mathrm{Mod}$$T \otimes_R (-) \colon R Mod \to T Mod

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 universal localization or Cohn localization of the ring $R$ at $S$.

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.

Related concepts and literature

• localization of a ring
• (NLOC) Noncommutative localization in algebra and topology, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002), London Math. Soc. Lecture Notes Series 330 (pdf), ed. Andrew Ranicki, Cambridge University Press (2006)
• Z. Škoda, Noncommutative localization in noncommutative geometry, in (NLOC, above) pp. 220–313, math.QA/0403276
• Andrew Ranicki, Noncommutative localization in algebra and topology, talk at Knot theory meeting, 2008, slides pdf; Noncommutative localization, Pierre Vogel 65th birthday conference, Paris, 27 October 2010, slides pdf
• P. M. Cohn, Inversive localization in noetherian rings, Communications on Pure and Applied Mathematics 26:5-6, pp. 679–691, 1973 doi
• V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization, pdf

One can also look at localization with inverses just from one side:

• P. M. Cohn, One-sided localization in rings, J. Pure Appl. Algebra 88 (1993), no. 1-3, 37–42