localization of a module

Localization of a module is closely related to localization of a ring.

Definition

Let be an -module and be a multiplicative subset of . Similarly to the localization of , , the localized -module (denoted ) is the module where is the equivalence relationship The action on is defined as

Compatibility with

For the localization of , there is a canonical ring homomorphism

Similarly, we have a map such that . If we consider to be an -module then we would like this map to be an -module homomorphism. Note that, Therefore, is a morphism when we restrict ourselves to looking at inside .

Properties

where tensoring over R.

Proof

Consider the bilinear map:

Then by the universal property of tensor products, with the bilinear map,

we have a unique linear map, such that the following diagram commutes:

This means we have the linear map,

as is linear, we can define it only on pure tensors. Next consider the function,

Given two representatives , then

thus, is well-defined.

Note is linear since,

The action by an element can clearly be factored out as well.

Therefore, since we have,

then , and by a similar computation .

Therefore, .

As a functor

Localization defines an exact functor

Proof

First, let’s be very explicit about what the functor is defined to be.

Consider the following short exact sequence,

We know that the functor is right exact so we have another exact sequence,

By the isomorphism above, we have an exact sequence,

Therefore, it suffices to check that given an injective map , that the corresponding map is injective.

Let , hence,

As is injective, this implies that and is injective so the following diagram is exact:

Therefore, is an exact functor.