Derived localization of algebras and modules

Localization of commutative rings and modules is a basic procedure in commutative algebra;
one of its main properties is exactness. This property allows easily to define the corresponding
notion on the level of homotopy categories. Noncommutative localization is much harder, but
also much more interesting. It comes up in various contexts, e.g. the construction of a derived
category could be viewed as localization with respect to quasi-isomorphisms.

 I will explain how to construct localization of noncommutative algebras and modules over
them so that exactness is preserved. Among the applications are a general form of the group
completion theorem, cyclic and graph homology, idempotent ideals etc.
 
This is joint work with J. Chuang and C. Braun.