nLab
universal localization

Idea

The notion universal localization or 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.

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 $New = \left(\array{ A & 0\\ C & B}\right)$, matrix $New$ is also in $\Sigma$.

We say that a homomorphism 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 $R/Ring$). 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 $A\in\Sigma$.

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 $f$

$$T \otimes_R (-) \colon R Mod \to T Mod$$

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.

References

The existence of the universal localization is exhibited in

• P. M. Cohn, Free rings and their relations, Academic Press 1971

Original articles include

• P. M. Cohn, Inversive localization in noetherian rings, Communications on Pure and Applied Mathematics 26:5-6, pp. 679–691, 1973 doi

Reviews and lecture notes include

• V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization, pdf

• (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

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

Universal localization of group rings (and connections to certain noncommutative rational function rings and Fox derivatives) is discussed in

• M. Farber, Pierre Vogel, The Cohn localization of the free group ring, Math. Proc. Camb. Phil. Soc. (1992) 111, 433 (pdf)