nLab
universal localization

Contents

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 1×11 \times 1-matrices) but forces more general matrices with coefficients in the ring to become invertible.

Definition

Let Σ\Sigma be a multiplicative set of matrices over a (typically noncommutative) ring RR.

We say that a homomorphism of rings f:RSf: R\to S is Σ\Sigma-inverting if all matrices f(A)f(A) over SS where AΣA\in \Sigma are invertible in SS. The Cohn localization of a ring RR, is a homomorphism of rings RΣ 1RR\to \Sigma^{-1} R which is initial in the category of all Σ\Sigma-inverting maps (which is the subcategory of coslice category R/RingR/Ring).

More general definition

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

T R():RModTMod T \otimes_R (-) \colon R Mod \to T Mod

sends all morphisms of SS into isomorphism in the category of left TT-modules.

P. M. Cohn has shown that there is a universal object RQ SRR\to Q_S R in the category of SS-inverting morphisms. The ring Q SRQ_S R (and more precisely the universal morphism itself) are called the universal localization or Cohn localization of the ring RR at SS.

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

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 is discussed in

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

Revised on December 6, 2013 23:08:29 by Tim Porter (2.26.40.111)