Direct Limit of Modules (Pt. I)
Point of Post: In this post we discuss the notion of the direct limit of modules, giving some particular examples of direct limits.
In this post we shall discuss one of the most useful constructions in the entirety of module theory–direct limits. Intuitively direct limits allow us to define a gluing process which makes rigorous sense of things such as the following: “ is the limit of the set of all finite products of “. To be more specific we wish to look at the case when we have a “chain” (really, we shall be discussing a more general notion, but chain gives the right idea) of modules and a chain of morphisms
and we wish to glue the chain upwards in a way the respects the morphisms–in other words we’d like to, at least intuitively, glue them to get some object for which there are always maps which respect the mappings with in the sense that should be equal to . Thus, in a sense we are really taking a limit of the , allowing us to often realize certain objects as the limit of certain finitary objects. We shall end up seeing that this limit is a fairly faithful representation of the individual in the sense that coproducts are good representation of the factor modules–this shouldn’t be surprising since coproducts are themselves direct limits. All in all, the intuitive idea of direct limits are that they are a two-step process consisting of gluing a set of modules together and then identifying the elements of the gluing which are “eventually equal” (in the sense of the limit).
Directed Sets and Directed Systems
We define a directed set to be a pair where is some preorder with the property that any two elements of possess an upper bound in (i.e. for any there exists some such that and ). Directed sets can be seen as a generalization of total orderings (i.e. chains) since while not every two elements of are related, we see that there is some interaction between them. For example, given a set the poset is not totally ordered in general (in fact, it’s only totally ordered if is a singleton) but it is a directed set since is always an upper bound for any two elements of .
Given a preordered set (most often times, it shall be a directed set) a directed system of module over is a set of left -modules and a set of -maps such that and . Intuitively, this second condition means that if we take the natural directed graph for (i.e. the directed graph with vertices and a directed edge if and only if ) and replace each vertex by the corresponding module and each directed line and replace it with the connecting morphisms one gets a commutative diagram.
Let’s take a look at some examples of directed systems of modules. Take a ring and consider the -modules for each . We have the obvious inclusions for given by . Clearly then and so that and forms a directed system of modules over the directed system (in fact, totally ordered set) .
Any set of modules can be made, in a trivial way, into a directed system by defining the trivial preorder on (i.e. and nothing else is comparable) and defining .
Consider the following slightly more interesting example. Let be some topological space and fix some . Let denote the neighborhood system of and consider for each the -algebra of continuous maps . Define the obvious preorder on by if (so the for those who have done any category theory or order theory) . This is clearly a directed set since given any two we know that and . Consider now the restriction homomorphisms given by (i.e. restriction of a mapping on to a mapping on ). It’s not hard to see then that and forms a directed system over the directed set .
As a last example consider for a second some fixed -module and let denote the set of all finitely generated submodules of . Clearly is a directed set. Moreover, it’s clear that if we define for the inclusion maps then it’s not hard to see that and is a directed system over .
Now that we have successfully defined the directed system of modules we can define direct limits over such directed systems. Indeed, given a preordered set and a directed system consisting of modules and morphisms we define, if one exists, a direct limit of the system to be an ordered pair where is a module and a a set, called a limit cone, of -maps such that whenever and which satisfies the following universal property: given morphism (where is some left -module) such that then there exists a unique morphisms such that .
If we have a preordered set and a directed system consisting of and we may denote that a module is a direct limit of this directed system by writing , or when no confusion will arise, .
This object can be thought of as a natural way to complete a particular diagram. Indeed, as was mentioned above one can think of getting a certain commutative diagram from our directed system by replacing the vertices in the natural directed graph induced by by their associated modules , and given any edge replacing it by the connecting homomorphism . We see then that is precisely the “outward mapping completion” of in the sense that if we add to the module and the (mapping out of !) elements of the limit cone we get a new diagram which is still commutative.
As usual with these universal constructions we have that direct limits, if they exists, are unique up to unique isomorphism. Indeed:
Theorem: Let be preordered set and with a directed system. Then, if and are two direct limits of this direct system then .
Proof: Since are a set of maps such that we know by the universal property of that there exists a unique map such that . Similarly, we are afforded a map such that . We see then that and . Now, since clearly the identities and clearly satisfy the respective properties we have by uniqueness that and from where it follows that is an isomorphism.
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 Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.
 Blyth, T. S. Module Theory. Clarendon, 1990. Print.
 Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.