nLab
universal localization

Contents

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 set of finite square matrices (of different sizes) over a (typically noncommutative) ring RR. Without loss of generality, one assumes that Σ\Sigma is left or right multiplicative. It is left multiplicative if for any matrices A,B,CA,B,C of right sizes such that A,BΣA,B\in\Sigma and CC fits into matrix New=(A 0 C B)New = \left(\array{ A & 0\\ C & B}\right), matrix NewNew is also in Σ\Sigma.

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). In the left hand version, the elements in the localized ring are thought of as solutions of linear equations Ax=bA x = b where bb is a column vector with elements in RR and AΣA\in\Sigma.

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

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)

Last revised on November 2, 2019 at 12:03:50. See the history of this page for a list of all contributions to it.