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 $1 \times 1$-matrices) but forces more general matrices with coefficients in the ring to become invertible.
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 $A\in\Sigma$.
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 $f$
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 $S$.
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 existence of the universal localization is exhibited in
Original articles include
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:
Universal localization of group rings (and connections to certain noncommutative rational function rings and Fox derivatives) is discussed in
Last revised on November 2, 2019 at 12:03:50. See the history of this page for a list of all contributions to it.