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 -matrices) but forces more general matrices with coefficients in the ring to become invertible. One also considers the corresponding localization functor on the category of modules. It can be related to ( of) some Bousfield localization (on chain complexes of modules).
Let be a set of finite square matrices (of different sizes) over a (typically noncommutative) ring . Without loss of generality, one assumes that is left or right multiplicative. It is left multiplicative if for any matrices of right sizes such that and fits into matrix , matrix is also in .
We say that a homomorphism of rings is -inverting if all matrices over where are invertible in . The Cohn localization of a ring , is a homomorphism of rings which is initial in the category of all -inverting maps (which is the subcategory of coslice category ). In the left hand version, the elements in the localized ring are thought of as solutions of linear equations where is a column vector with elements in and .
Given a ring and a family of morphisms in the category Mod of (say left) finitely generated projective -modules, we say that a morphism of rings is -inverting if the extension of scalars from to along
sends all morphisms of into isomorphism in the category of left -modules.
P. M. Cohn has shown that there is a universal object in the category of -inverting morphisms. The ring (and more precisely the universal morphism itself) are called the universal localization or Cohn localization of the ring at .
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.
The original articles:
Paul M. Cohn, Free rings and their relations, Academic Press (1971) [pdf]
Paul M. Cohn, Inversive localization in noetherian rings, Communications on Pure and Applied Mathematics 26 5-6 (1973 ) 679-691 [doi:10.1002/cpa.3160260510]
Reviews and lecture notes:
V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization (2007) [pdf]
Andrew Ranicki (ed.), Noncommutative localization in algebra and topology, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002) London Math. Soc. Lecture Notes Series 330 Cambridge University Press (2006) [pdf]
containing this article on applications to topology:
reviewed in:
Andrew Ranicki, Noncommutative localization in algebra and topology, talk at Knot theory meeting (2008) [slides pdf]
Andrew Ranicki, Noncommutative localization, Pierre Vogel 65th birthday conference, Paris, 27 October 2010 [slides pdf]
See also:
On localization with inverses just from one side:
Universal localization of group rings (and connections to certain noncommutative rational function rings and Fox derivatives) is discussed in
Examples for commutative rings
Last revised on October 8, 2024 at 09:14:32. See the history of this page for a list of all contributions to it.