A Different Way of Looking at Direct Limits
Point of Post: In this post we discuss a perhaps more intuitive way to think about direct limits.
So, we have already discussed what it means to have a “directed system” of modules and what a “direct limit” of this directed system may look like. To recant, we can think of a directed system of modules as being a “lattice” (I mean this only in the visualization sense) diagram of modules with the property that whenever one module “sits below” another module there is a map . Intuitively, we thought about these maps as being embeddings, so what we could view as sitting inside (while this is true most of the time, we see that there is no actual need to require these maps be embeddings). What we were then curious about (especially in the case that our indexing set was directed) was what the “limiting behavior” of this diagram looked like. Namely, while the diagram may not have a “highest member” (i.e. while our indexing set might not have a largest member, so that there is a module sitting at the “top” of the diagram) we see that, at least intuitively, going up the diagram gives us modules that are getting closer together. Indeed, assume for a second that our indexing set is directed. Then, at least intuitively going high enough allows one to put any two modules “close together”, since if we are dealing with by assumption that our preordered set is directed we can find some and so and putting them “close together”. Thus, we see that going up the lattice diagram gives us modules that seem to be “approximating” some module, but most often this module isn’t part of the diagram–the direct limit is this module, the ideal “top module” which ‘should’ be there. One can completely analogize this situation to completing a metric space so that all Cauchy sequences converge. To make the translation more apparent, we add in the direct limit to our lattice diagram to “complete it”, so that that our “Cauchy sequence” of modules actually has a limit–the direct limit.
That said, even with all of this intuition about what the direct limit “does” it’s hard to get a firm grasp on what it should “look like”. That is the goal of this post, to describe direct limits over directed systems of modules (over directed sets, with injective put-in maps) as “directed unions”. What are directed unions? And, why do they help us understand direct limits? Well, directed unions are just, in essence, unions of submodules, but the collection of submodules (over which we are unioning) is such that we actually get a submodule. I think this will be more clear if one recalls that for two submodules one has that is a submodule of if and only if, without loss of generality, . Thus, we shall see that having a set of submodules of some fixed module is exactly the condition one needs so that is actually a submodule of –moreover (and we have actually already proven this) the union is precisely the direct limit of the directed system of submodules! Now, this construction is easy to picture, it’s just a union. The point of this post shall be to show that (for sufficiently nice examples) all direct limits are of this form! Roughly, once we put our directed system of modules “inside the same module” we will see that the direct limit of this system really is just the directed union of these submodules.
Direct Limits as Directed Unions
We begin by defining precisely what we mean by a directed system of submodules and what we mean by their directed union. In particular, let be some ring and some fixed left -module. We then define a set of submodules of to be directed if we can find a directed set such that may be indexed as such that it becomes a directed system over with the inclusion maps . Or, said more concretely, will be directed if and only if given any one can find some with . If is a directed system of submodules we call the union over the directed union of . What we claim now is the following:
Theorem: Let be a fixed module. Then, a directed set of submodules of has the property that its union is also a submodule.
Proof: Assume first that is directed and let denote the union over . Let and be arbitrary. To see that merely note that for some and since one has that . To see that note that by assumption we can find with and . But, since is directed we can find some such that and since this implies that . It follows that as desired.
For what comes next we need to define the general notion of a cone for a directed system. Namely, given a directed system over some preordered set and some module we say a cone is a set of maps with the property that . Thus, we see that limit cones are special cases of cones. The following theorem characterizes when a cone is, in fact, a limit cone:
Theorem: Let be a directed set and a directed system over . Then, a cone is a limit cone if and only if the following conditions are satisfies the following conditions
If is a limit cone then we further have that if and only if for some .
Proof: Suppose first that is a limit cone. We begin by noting that is a directed set of submodules of . To see this we merely need to show that given any we can find some that contains them both. To see this, we merely note that since is directed we can find a such that . To see that this implies we merely note that by assumption that is a cone that for any and we have that and from where the desired inclusions clearly follow. From this, and the above theorem concerning directed sets of submodules, we know that the union over , call it , is a submodule of . Our goal now is to show that . To do this note that we have the obvious maps which evidently satisfy the compatibility requirement, and since is a limit cone we know we get a map such that . Note then that if we consider the inclusion we have that for all , but since also satisfies this property, by uniqueness we may conclude that . But, this implies that, as a map, has a right inverse, and thus is surjetive, or equivalently . Now, fix some , we wish to show holds. But, note that since is a direct limit of this directed system we have a natural isomorphism , which is such that it suffices to prove this result holds for $latex\ varinjlim M_\alpha$, but this is trivial.
Conversely, suppose that is a cone satisfying these two properties. Suppose we are given a set of maps (where is some fixed left -module) such that . Define a map by taking to where (by we know that we can find such a representation). To see that this map is well-defined suppose that as well. Since is directed we can choose and so we have by assumption that or . But, by this allows us to choose such that or . Thus,
and so is well-defined. Note as well that for any and so that satisfies the compatibility relations. It’s evident that this is unique by since if is another map satisfying the compatibility relations then , and since every element of is of the form it follows that . Thus, is a limit cone as claimed.
This last claimed property was proven to be true (in the previous paragraph) if the cone satisfies and since (by the first paragraph) all limit cones satisfy these properties this follows.
This gives us the satisfying result we’ve been looking for. Namely, if we assume that each is an embedding (as is most natural) then given any direct limit (where is directed) we see by above that each is an embedding and moreover from the theorem we also see that is the just the direct union of the . Thus, direct limits in this case are nothing more than just direct unions where the “factors” had the unfortunate turn of fate to not be born into the same module!
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