Algebras and modules
Model category presentations
Geometry on formal duals of algebras
A localization of a module is the result of application of an additive localization functor on a category of modules over some ring .
When is a commutative ring of functions, and under the interpretation of modules as generalized vector bundles the localization of a module corresponds to the restriction of the bundle to a subspace of its base space.
For a (possibly noncommutative) unital ring, let Mod be the category -modules. Here may be the structure sheaf of some ringed topos and accordingly the modules may be sheaves of modules.
Consider a reflective localization functor
with right adjoint . The application of this functor to a module is some object in the localized category , which is up to isomorphism determined by its image .
The localization map is the component of the unit of the adjunction (usually denoted by , or in this setup) .
Depending on an author or a context, we say that a (reflective) localization functor of category of modules is flat if either is also left exact functor, or more strongly that the composed endofunctor is exact. For example, Gabriel localization is flat in the first, weak sense, and left or right Ore localization is flat in the second, stronger, sense.
By the Eilenberg-Watts theorem, if Mod then the localization of a module
is given by forming the tensor product of modules with the localizatin of the ring , regarded as a module over itself.
If the localization is a left Ore localization or commutative localization at a set then is the localization of the ring itself and hence in this case the localization of the module
is given by extension of scalars along the localization map of the ring itself.
In these cases there are also direct constructions of (not using to ) which give an isomorphic result, also denoted by .
Standard discussion over commutative rings is for instance in
- Andreas Gathmann, Localization (pdf)
Discussion in the general case of noncommutative geometry is in
- Z. Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276