Localization of commutative rings may be characterized by a unversal property whose morphisms are contained in the category of commutative rings. For modules, it is harder to formalize precisely since localizing changes the base ring, i.e. moves a module and its morphisms to a different category. It is the same phenomenon one encounters when dealing with the universal property characterizing free objects.
Given two functors , we can form the comma category, denoted , as follows:
- objects are triples where is an object of , is an object of , and is an arrow .
- arrows from to are pairs where is an arrow and is an arrow , such that the following diagram commutes:
Composition is defined by concatenation of commutative diagrams, which is clearly associative, and the identity arrow for the object is simply .
As a particular case, if is a functor , we identify with the single object to which it maps. An object of is called an arrow from to . A universal arrow from to is then simply an initial object in the category .
Obviously, the data for an arrow from to may be compressed to a pair instead of a triple. Similarily, the data for an arrow between and may be more simply given as a single arrow making the appropriate diagram commute. Thus we can describe more explicitely a universal arrow from to :
- it is a pair with an object of and an arrow
- such that for any other pair with
- there exists a unique in such that
Because initial objects are unique up to a unique isomorphism, so are universal arrows.
We fix once and for all a commutative ring with unit , together with a multiplicative submonoid .
Let be the functor defined by restriction of scalars along the canonical ring homomorphism . Now, given any -module , its localization at is the universal arrow from to . We write this universal arrow as and often call it the “canonical localization morphism”. By definition it is an -linear map.
As with any characterization via a universal property, we get uniqueness for free but we need to work a bit to show existence. We will make an elementary construction that realizes the stated universal property. There are many ways to do it. We set and we define the -linear map to be the obvious, canonical one given by the tensor product. The -module structure on is given by .
Now suppose is some -module, and that we have an -linear map . Now, we may define a map given by , which is obviously -bilinear. Then the universal property of the tensor product produces a canonical -linear map , the unique one such that the following diagram commutes: Given and an elementary tensor , so that is actually an -linear map, and moreover this fact only depends on the previous diagram commuting and the -module structure on . We see that is a morphism between and in the category , and it makes the following diagram commute in :
Let’s tackle unicity. Suppose and are two -linear maps such that we have . We will show that in that case . To achieve this goal, it suffices to show and agree on elementary tensors . This is easy: The construction is complete.
We let be the full subcategory of whose objects are -modules such that, for all , scalar multiplication by is an automorphism (i.e. all have “invertible action” in the module). We can actually exhibit an isomorphism of categories between and . They really are the same thing, in a strong sense.
I won’t do the details of the isomorphism, but one interesting question is, how to put an -module structure on some object of ? The answer is: given some fraction in and some element , there exists some such that ; we define