By Andrew Ranicki

Noncommutative localization is a strong algebraic procedure for developing new jewelry by way of inverting components, matrices and extra more often than not morphisms of modules. initially conceived by way of algebraists (notably P. M. Cohn), it's now an incredible instrument not just in natural algebra but in addition within the topology of non-simply-connected areas, algebraic geometry and noncommutative geometry. This quantity contains nine articles on noncommutative localization in algebra and topology by way of J. A. Beachy, P. M. Cohn, W. G. Dwyer, P. A. Linnell, A. Neeman, A. A. Ranicki, H. Reich, D. Sheiham and Z. Skoda. The articles comprise uncomplicated definitions, surveys, old heritage and functions, in addition to proposing new effects. The publication is an creation to the topic, an account of the state-of-the-art, and likewise offers many references for extra fabric. it truly is compatible for graduate scholars and extra complex researchers in either algebra and topology.

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**Example text**

In other words, Y is f -local if, as far as mapping into Y is concerned, f looks like an equivalence. 2 Definition. A map X → X of R-modules is said to be an f -local equivalence if it induces an equivalence HomR (X , Y ) → HomR (X, Y ) for every f -local R-module Y . An f -localization of X is a map : X → Lf (X), such that Lf (X) is f -local and is an f -local equivalence. 3 Remark. It is not hard to see that any two f -localizations of X are equivalent, so that we can speak loosely of the f -localization of X.

If Lf is smashing then the category of f -local R-modules is equivalent, from a homotopy point of view, to the category of Lf (R)modules. In particular, the homotopy category of f -local R-modules is equivalent to the homotopy category of Lf (R)-modules. 9 Examples. Let R = Z, pick a prime p, and let f be the map Z − → Z. Then Lf is smashing, and Lf (X) ∼ Z[1/p] ⊗Z X. 5], which is the total left derived functor of the p-completion functor. In particular, Lf (Z) ∼ Zp and Lf (Z/p∞ ) ∼ ΣZp . Since ΣZp is not equivalent to Zp ⊗Z Z/p∞ , Lf is not smashing in this case.

More generally, the homotopy category of any Quillen model category [6] [11] can be built by formally inverting maps. In a slightly different direction, the process of localization with respect to a map (§2) has recently developed into a powerful tool for making homotopy-theoretic constructions [2, §4]; roughly speaking, localizing with respect to f involves converting an object X into a new one, Lf (X), with the property that, as far as mapping into Lf (X) goes, f looks like an equivalence. 2).

### Noncommutative localization in algebra and topology by Andrew Ranicki

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