localization of a module
if and only if there is some such that
on is an equivalence relation.
Clearly as for any , where . Also, implies that , since implies that . Finally, given and , we are led to two equations and for some . Expanding and rearranging these, then multiplying the first equation by and the second by , we get . Since , as required. ∎
Let be the set of equivalence classes in under . For each , write
the equivalence class in containing . Next,
define a binary operation on as follows:
together with and defined above is a unital module over .
That and are well-defined is based on the following: if , then
which are clear by Proposition . Furthermore is commutative and associative and that distributes over on both sides, which are all properties inherited from . Next, is the additive identity in and is the additive inverse of . So is a module over . Finally, since for any , so that is unital. ∎
Definition. , as an -module, is called the localization of at . is also written .
The notion of the localization of a module generalizes that of a ring in the sense that is the localization of at as an -module.
If , where is a prime ideal in , then is usually written .
|Title||localization of a module|
|Date of creation||2013-03-22 17:26:59|
|Last modified on||2013-03-22 17:26:59|
|Last modified by||CWoo (3771)|