symmetric monoidal (∞,1)-category of spectra
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.
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 ).
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.
Original articles include
Reviews and lecture notes include
(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)
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:
Universal localization of group rings is discussed in